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f(^X^0*f v ~$Y~(fWX~8YV0\$^\$^f($^fWY\N8^~$ ($$%׃ $D$$$DŽ_ s$4* $݄$^݄$^@n YnXn0h ^(Y^X^8XV݄$݄$ɋ$D$\$ \$$ʠ$$$u&݄$݄$$u&݄$v݄$2$DŽ$$9$ $T$X<΋$$t$XɃ$ ti$pG$tG$xG $|G$G$G$G$G $G$($$GD$\$G$G $G$G$G$G$G $G$($$i$%@te$$$ 0D$\T$D$ $D$$pD$$$$$9$`x L$4ϋ($y$,$$tl$$$$D$ |$T$4$$4$#$9$t$$$$Ɖ+(ve7$G"݄$$F݄$$nF݄$$\F ݄$$JF[^_]$V덍$$$D$ l$L$$3w$$$D$JD$\D$ $$D$$pD$D$4$ ${D$$$${$D$ $$t$\D$$pt$ D$$$胠%$D$$臩$DŽs$4$芶݄$^FF@V YVXV0P N(YNXN8Hu:VZ݄$$D$\$ \$$FV$l$O@T$D$ T$$($D$$M$DŽs$4>$PFF݄$^@~ Y~X~0x v(YvXv8pu:V݄$Z$D$\$ d$$Ɣ}FV$L$@D$ T$t$$*݄$݄$ɋ$T$0\$ ݄$T$(T$\$\$D$t$8,$ ݄$݄$T$0\$ $݄$ɉt$8T$(T$\$\$D$,$ X݄$݄$ɉ\$0݄$ɋ$\$(T$ ݄$\$݄$\$\$D$t$8,$i ݄$݄$ɋ$\$0݄$\$(T$ ݄$\$݄$ɉt$8\$\$D$,$ FV<$T$_D$ D$L$$8z$Ux1$l$7D$ l$t$$$D$ D$T$$FV]$l$$D$ D$t$$[FV!$L$3$D$ D$|$$[FV$T$$D$ D$|$$@FV$T$mD$ L$l$$`,>o+r.G"$&UWVS6l$t$T$у1VUUU)4RT$L9Yt$L4$?4$D$X34$D$T'D$P$D$\D$L9D$\i#l$@ED$Hu} t?$T$9$tL$@4$L$@t$@4$_Z\$(*L$(YX,‰D$DtD$@<$D$|$@<${cZ\$(*t$(YX,ċ|$\L$DT$T t$P1|$HGt+%L$Hqщ څNT$\D$X 4D$LT$\9$t[$L$LD$Pt$Tl$X|$L$ D$t$,$ǼL$T $D$P$t$X4$l[^_]ËD$PT$Ll$TD$D$XT$ l$$m뮍D$ t$|$$[,$J ?$T$!F1҉L$XVUUU,f(剄$Y,Y)$$UWVSl$D$,<ȉ͉L$X+)9@D$,(%D$P "D$LEl$TL$,+)9Ɖt$,>D$PD$L|$H$0 L$D\;$$L$,h+l$@D$@t$,D$\ŋD$\|$\t$@D|$\~ԋT$,2~+D$,8F"$19$GL$4>T$0D$,D$TD$H$D$8D$D$ GT$TT$,$D$/AD$D$ G,$D$D$HD$AD$09D$D$D$D$t[ED$ uKED$D$DD$$$T$89D$,|$XD$,8l[^_]ËE볋E D$D$ L$4l$Hl$D$D$T$;l$T=$茲ZT$\~Ot$,>w+9w"D$,0GF@T$\V$2ЋD$\؉D$= 0, found amax=%Z ratlift: bmax must be > 0, found bmax=%Z invmodbezoutNaN or Infinity in dbltorrtodbloverflow in real shiftFlx_to_Flvnon invertible polynomial in Flxq_invFlxqX_safegcddiscriminant not congruent to 0,1 mod 4 in %ssquare discriminant in %szero discriminant in %snegative discriminant in %spositive discriminant in %squadpolynegative definite t_QFIShanks distance must be a t_REAL in qfrQfbzero discriminant in Qfbdifferent discriminants in qfb_compcompositionqfr_unitqfr_unit_by_discnot a t_REAL in 4th component of a t_QFRqfi_unitqfi_unit_by_discnot a t_QFR in powrealrawnot a t_QFI in powimagnot a t_QFI in nucompnot a t_QFI in nuduplnot an integer exponent in nupowredimagsl2redimagreducible form in qfr_rhonot a real quadratic form in redrealreducible form in qfr_initqfbreddiscriminant not congruent to 0,1 mod 4 in primeformqfbsolvecornacchiad must be positived must be 0 or 3 mod 4RgX_RgXQ_compoRgX_powersnormalizing a polynomial with 0 leading termshallowtransgtransimpossible concatenation: %s %Z . %s %Ztrying to concat elements of an empty vectorconcatno such component in vecextractincorrect mask in vecextractincorrect range in extractmask too large in vecextractextractmatextractincorrect length in sumnegative size in fill_scalmatnegative size in fill_scalcolisdiagonalincorrect object in diagonalmatmuldiagonalincorrect vector in matmuldiagonalmatmultodiagonalmattodiagonalgaddmathnfdividehnf_invimageSolving the triangular system gauss. i=%ldEntering gauss with inexact=%ld negative size in matid_FlmFpM_gauss. i=%ldZM_invZM_inv doneinverse mod %ld (stable=%ld)detintnot an integer matrix in detintdetint. k=%ldkerigauss_pivot_ker. k=%ld, n=%lddeplinempty matrix in deplininverseimageimage2matimage* [mod p]FpM_gauss_pivotFqM_gauss_pivoteigenmissing eigenspace. Compute the matrix to higher accuracy, then restart eigen at the current precisionmatdetdet2detdet, col %ld / %lddet. col = %ldgaussgauss_pivot. k=%ld, n=%ldgauss_pivotempty matrix in supplFpM_kerFpM_invimageFqM_kerFlxqM_kergaussmodulogauss_pivot_kercharpolyincorrect variable in caradjminpolyhesshess, m = %ldgnormgnorml2gnorml1QuickNormL1gconjnot a rational polynomial in conjvecconjvecincompatible field degrees in conjvecassmatgtracesqred1not a positive definite matrix in sqred1sqred3jacobimatrixqz0matrixqznot a rational matrix in matrixqzmatrixqz when the first 2 dets are zeromatrix of non-maximal rank in matrixqzmore rows than columns in matrixqzmatrixqz2matrixqz3intersectmathnfhnf_specialincompatible matrices in hnf_specialhnf_special[2]. i=%ldhnf_special[1]. i=%ldhnfspec [%ld x %ld] --> [%ld x %ld] matb cleaned up (using Id block) hnfspec[3], (i,j) = %ld,%ld after phase2: Permutation: %Z matgen = %Z hnfspec[2]hnfspec[1] after phase1: Entering hnfspec mathnfspec with large entrieshnfadd (%ld + %ld)H = %Z C = %Z 2nd phase done 1st phase done non coprime ideals in hnfmergehnf[2]. i=%ldhnf[1]. i=%ldhnflllhnflll, k = %ld / %ldhnflll (reducing), i = %ldextendedgcdhnfpermhnfall hnfall[3], j = %ld hnfall, final phase: hnfall[2], li = %ldhnfall[1], li = %ldsmithallnon integral matrix in smithall[3]: smithall[2]: smithall, i = %ld; [1]: smithall i = %ld i = %ld: starting SNF loopsmithcleanmatsnfaccuracy lost in matfrobeniusmatfrobeniusvariable must have higher priority in matfrobeniuseasycharsqred2matrixqz_auxLeaving hnffinal mit = %Z B = %Z C = %Z dep = %Z first pass in hnffinal done hnffinal, i = %ld hnflll done Entering hnffinal: mit = %Z B = %Z allhnfmodallhnfmod[2]. i=%ldallhnfmod[1]. i=%ldnb lines > nb columns in hnfmodgsmithallassociationnot an element of (Z/nZ)* in orderprimitive root mod %Z does not existprimitive root mod 2^%ld does not existzero modulus in znprimrootnegative integer in sqrtintZ_issquareispowerispower for non-rational argumentsmissing exponentisanypowerisanypower: now k=%ld, x=%Z insufficient precision for p = 2 in hilbertforbidden or incompatible types in hilcomposite modulus in Fl_sqrt: %lucomposite modulus in Fp_sqrt: %ZFp_sqrtnot a prime in Fp_sqrtFp_sqrtn1/0 exponent in Fp_sqrtnchineselarge exponent in Mod(a,N)^n: reduce n mod phi(N)gisprimenegative argument in factorial functionsfcontintegral part not significant in sfcontnegative nmax in sfcontcontfrac0incorrect size in pnqnpnqnbestappr0bestapprincorrect bound type in bestapprfundunitregulaexponent overflow in regulaqfbclassnoclassnoclassno with too small orderdiscriminant too big in classnoclassno2discriminant too large in classnohclassnoFp_sqrtlsfcont2list of numerators too short in sfcontf2n-th prime meaningless if n = %ldprimepipanic: set_optimizeToo large primelimitaddprimecan't accept 0 in addprimesremoveprimeprime %Z is not in primetablezero argument in an arithmetic functionissquarefreetoo many divisors (more than %ld)denominators not allowed in divisorsdivisorsbinarybitwise negationnegative exponent in bitwise negationbitwise orbitwise andbitwise xorbitwise negated implyunsupported type of cache modelIFAC: (Partial fact.) Initial stop requested. zero argument in factorintIFAC: Stop: remaining %Z IFAC: Stop: Primary factor: %Z checkrnfcheckbnfplease apply bnfinit firstchecknfplease apply nfinit firstincorrect bigray fieldplease apply bnrinit(,,1) and not bnrinit(,)missing units in %sincorrect idealincorrect matrix for idealincorrect bigidealincorrect prime idealpolynomial not in Z[X] in %spolynomial not in Z[X,Y] in %sincompatible modulus in %s: mod = %Z, nf = %ZtschirnhausTschirnhaus transform. New pol: %Zgpolcomp (different degrees)S1S2A3S3C(4) = 4E(4) = 2[x]2D(4)A4S4C(5) = 5D(5) = 5:2F(5) = 5:4A5S5C(6) = 6 = 3[x]2D_6(6) = [3]2D(6) = S(3)[x]2A_4(6) = [2^2]3F_18(6) = [3^2]2 = 3 wr 22A_4(6) = [2^3]3 = 2 wr 3S_4(6d) = [2^2]S(3)S_4(6c) = 1/2[2^3]S(3)F_18(6):2 = [1/2.S(3)^2]2F_36(6) = 1/2[S(3)^2]22S_4(6) = [2^3]S(3) = 2 wr S(3)L(6) = PSL(2,5) = A_5(6)F_36(6):2 = [S(3)^2]2 = S(3) wr 2L(6):2 = PGL(2,5) = S_5(6)A6S6C(7) = 7D(7) = 7:2F_21(7) = 7:3F_42(7) = 7:6L(7) = L(3,2)A7S7galoisgalois of reducible polynomialgalois (bug1)galois (bug4)galois (bug2)galois (bug3)galois of degree higher than 11incorrect galois automorphism in galoisapplygaloisapplyget_bnfpolfalse nf in nf_get_r1false nf in nf_get_r2false nf in nf_get_signmatricesmult. tablenfinitround4LLL basispolrednon-monic polynomial. Result of the form [nf,c]incorrect nf in nfnewpreci = %ld polred for non-monic polynomialordrednf_ADDZK flag when nf_ALL set (polredabs)Found %ld minimal polynomials. rootsof1rootsof1 (bug1)not an integer type in dirzetaktoo many terms in dirzetakneed %Z coefficients in initzeta: computation impossible initzeta: N0 = %Z i0 = %ld Ciklog(n)a(n)coefa(i,j)not a zeta number field in zetakallgzetakalls = 0 is a pole (gzetakall)s = 1 is a pole (gzetakall)nsiso0nfiso or nfinclmatrix Mget_red_Gget_red_G: starting LLL, prec = %ld (%ld + %ld) xbest = %Z you found a counter-example to a conjecture, please report!polredabs0 generator: %Z get_polpolredabs (precision problem)chk_gen_init: new prec = %ld (initially %ld) chk_gen_init: skipfirst = %ld precision too low in chk_gen_initchk_gen_init: generator %Z chk_gen_init: difficult field, trying random elements chk_gen_init: subfield %Z allbaseResult for prime %Z is: %Z Treating p^k = %Z^%ld impossible inverse: %Zdisc. factorisationreducible polynomial in allbasenfbasis dedek: gcd has degree %ld initial parameters p=%Z, f=%Z new order: %Z entering Dedekind Basis with parameters p=%Z f = %Z, a = %Z --> %Z : IndexPartial: factor %Z^%ld IndexPartial: factorizationIndexPartial: discriminantget_normprimedecincorrect modpr formatmodpr initialized for integers only!nf_to_ffincorrect polynomial in rnf functionincorrect coeff in rnf functionrnf functionnon-monic relative polynomialsincorrect variable in rnf functionIdeals to consider: %Z^%ld not a pseudo-basis in nfsimplifybasisrnfdet2not a pseudo-matrix in rnfdetrnfsteinitznot a pseudo-matrix in %srnfbasisrnfisfreepolcompositum0compositumnot the same variable in compositumcompositum: %Z inseparablernfequationinseparable relative equation in rnfequationrnflllgram%ld k = rnfpolredrnfpolredabsreduced absolute generator: %Z absolute basisoriginal absolute generator: %Z this combination of flags in rnfpolredabsnot a factorisation in nfbasis leaving Decomp: f1 = %Z f2 = %Z e = %Z de= %Z bug in Decomp (not a factor), is p = %Z a prime? entering Decomp, parameters: %Z^%ld f = %Z (Fa, Ea) = (%ld,%ld) entering Nilord with parameters: %Z^%ld fx = %Z, gx = %Zprimedec: %Z is not primernfdedekindrnfordmax new order: %Z %Z pass no %ld treating %Z relative basis computed ordmaxROUND2: epsilon = %ld avma = %ld no root in nilord. Is p = %Z a prime?nilordnewtonsums: result doesn't fit in cache Increasing Fa (eq,er) = (%ld,%ld) ** switching to fast mode ** switching to normal mode beta = %Z rowred j=%ldnewtonsums fastnu: HNF(G) is computed fastnu: G is computed content in fastnu is %Z Increasing Ea nfmulelement_mulelement_invnfdivelement_divelement_mulielement_sqrnot an integer exponent in nfpowelement_pownegative power in element_powid_mod_pelement_mulidnot the same number field in basistoalgalgtobasis_iincompatible variables in algtobasisnot the same number field in algtobasisnot a matrix in matbasistoalgnot a matrix in matalgtobasisnot the same number field in rnfalgtobasisarch_to_permzero element in zsignezsigneelement_invmodidealmodule too large in Fp_shanksFp_shanks, k = %ldPohlig-Hellman: DL mod %Z^%ld nf_Pohlig-Hellman: DL mod %Z^%ld not an element of (Z/pZ)* in znlogIdealstar needs an integral non-zero ideal: %Zincorrect archimedean component in Idealstaridealstar%Z not a nfeltnot an element in zideallogideallistideallistarcha not invertible in ff_PHlog_Fpmodule too large in ffshanksffshankszidealij done treating a = %ld, b = %ld treating pr^%ld, pr = %Z leaving entering zlog, with a = %Z Ideallistincorrect ideal in idealtypidealhermiteprincipalidealget_archidealdivideal_two_eltzero ideal in idealfactoridealvalideals don't sum to Z_K in idealaddmultoonenot a vector of ideals in idealaddmultoonegeneric conversion to finite fieldidealnormidealinvcannot invert zero idealincompatible variables in idealinvnon-integral exponent in idealpownon-integral exponent in idealpowredquotient not integral in idealdivexactincorrect vector length in idealrednot a vector in idealredunif_mod_fZnored + denominator in idealapprfactnot a factorization in idealapprfactnot a prime ideal factorization in idealchinesenot a suitable vector of elements in idealchineseelement not in ideal in ideal_two_elt2element does not belong to ideal in ideal_two_elt2not a module in %snot a matrix in %snot a correct ideal list in %snfhermitenfhermite, i = %ldnot a matrix of maximal rank in nfhermitenot a module in nfsmithnot a matrix in nfsmithnot a correct ideal list in nfsmithnfsmithbug2 in nfsmithnfsmith for non square matricesnot a matrix of maximal rank in nfsmithnfkermodprnfsolvemodprnfdetintnfhermitemod[2]: nfhermitemod, i = %ld[1]: nfhermitemod, i = %ld0 in get_arch_realincorrect idele in idealaddtoone0th power in idealpowprime_specplease apply rnfequation(,,1)incorrect data in eltreltoabsrnfinitalgmain variable must be of higher priority in rnfinitalgelement is not in the base field in rnfelementdownrnfidealhermiternfidealabstorellllintlllint_markedlllint[2], kmax = %ldlllint[1], kmax = %ldK%ld lllfp[1] (%ld) K%ldlllfplllfp[2], kmax = %ldin precision %ld Checking LLL basis... Checking LLL basis lllfp[1], kmax = %ld Recomputing Gram-Schmidt, kmax = %ld %ld ...LLL reducing precision to %ld k =lllfp giving upcount_max = %ld lllfp (exact)dependent vectors in lllfpqflllqflllgramgramlllintpartialmid = %Znpass = %ld, red. last time = %ld, log_2(det) ~ %ld lllintpartialalltm1 = %ZzncoppersmithEntering LLL bitsize bound: %ld expected shvector bitsize: %ld Increasing dim, delta = %d t = %d Roots: %Z Candidate: %Z bitsize Norm: %ld bitsize bound: %ld Matrix to be reduced: %Z Init: trying delta = %d, t = %d Modified P: %Z zero polynomial forbiddennegative bound in zncoppersmithnegative accuracy in lindep2lindep2lindepqzer[%ld]=%ld pslqInitialization time = %ld pslqL2inconsistent primes in plindepnot a p-adic vector in plindephigher degree than expected in algdepnegative polynomial degree in algdepalgdep0matkerintqfminimfincke_pohstFincke-Pohst, final LLL: prec = %ld first LLL: prec = %ld dimension 0 in fincke_pohstlllgramallgendelta = %d, t = %d, cond = %lf bound too largetime [max,t12,loop,reds,fin] = [%ld, %ld, %ld, %ld, %ld] time for ct = %ld : %ld minim0not a definite form in minim0.minim0, rank>=%ld*negative number of vectors in minim0bound = 0 in minim2maximal number of vectors must be providedfinal sort & check... sorting... New bound: %Zsmallvectorssmallvectors looking for norm < %Z incrementalGSgenadding vector = %Z vector in new basis = %Z list = %Z base change matrix = legendrenegative degree in legendreargument must be positive in polcyclonot a series in laplacenegative valuation in laplacenot a series in convoldifferent variables in convolprecision<=0 in gprecpolrecipbinomialtwo abcissas are equal in polintnot vectors in polinterpolatedifferent lengths in polinterpolateno data in polinterpolatenot a set in setsearchnot a set in setunionnot a set in setintersectnot a set in setminusdirmuldoubling stack in dirmul not an invertible dirseries in dirdivinvalid bound in randomn too small (%ld) in numtopermpermtonumnot a vector in permtonumreverse polmod does not existnot a polmod in modreversegen_sortvecsortindex too large in vecsortnegative index in vecsortincorrect lextype in vecsortnot a polynomial of degree 2 in quadhilbertquadhilbert needs a fundamental discriminantform_to_idealquadraynot a polynomial of degree 2 in quadrayyquadray needs a fundamental discriminant *** Bach constant: %f sorry, couldn't deal with this field. PLEASE REPORT rel = %ld^%ld cglob = %ld. buchquadbe honestbe honest for primes from %ld to %ld #### Tentative class number: %Z KC = %ld, need %ld relations ...need %ld more relations *** Changing sub factor base Bach constant <= 0 in buchquadnarrow class groupincorrect parameters in quadclassunitquadhilbertimagproduct, error bits = %ldrootsp = %lu, q = %lu, e = %ld class number = %ldcomputeP2get_lambdalambda = %Z [%ld,%ld] quadray: looking for [a,b] != unit mod 2f [a,b] = factor baseFB = %Z subFBquad (%ld elt.)powsubFBquadgeneratorssmith/class group Time %s rel [#rel/#test = %ld/%ld]: %ld random %ldPqrf5_rho_powreal_relationsinitialregulator is zero. #### Tentative regulator: %Z quadhilbertimag (can't find p,q)quadhilbert (pq)[quadhilbert] incorrect values in pq: %luincorrect data in findquadnot a vector/matrix in cleanarchincorrect big number field is %Z End of PHASE 1. #%ld in factor base Norm(P) > Zimmert bound Testing P = %Z *** p = %lu is %Z **** Testing Different = %Z PHASE 1: check primes to Zimmert bound = %lu red_mod_unitszero ideal in isprincipalisprincipalinsufficient precision for generators, not givenisunitnot a factorization matrix in isunitnot an algebraic number in isunitcodeprimebnfnewprecbnfmakebnfclassunitbnfinitnon-monic polynomial. Change of variables discardedfundamental units too largebnfinit: %s%s, not giveninsufficient precision for fundamental unitsgetfu #### Computing fundamental units SPLIT: increasing factor base [%ld] # ideals tried = %ld precision too low for generators, not givenprecision too low for generators, e = %ldclassgroup generators #### Computing class group generators completing bnf (building cycgen)*%ld makematalcompleting bnf (building matal)classgroupallincorrect parameters in classgroupbuchall (%s)cleanarchcompute_R #### Looking for random relations LIMC = %ld, LIMC2 = %ld R1 = %ld, R2 = %ld D = %Z Bach constant <= 0 in buchinitalg & rootsof1sub factorbase (%ld elements)powFBgenComputing powers for subFB: %Z ########## FACTORBASE ########## KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld ++ LV[%ld] = %Zisprincipal (incompatible bnf generators)BOUND = %.4g small norm relations small norms gave %ld relations. nb. fact./nb. small norm = %ld/%ld = %.3f for this idealv[%ld]=%.4g small_norm (precision too low) *** Ideal no %ld: [%Z, %Z, %Z, %Z] #### Looking for %ld relations (small norms) (more relations needed: %ld) looking hard for %Z ++++ cglob = %ld: new relation (need %ld)for this relationrelation cancelled: (jid=%ld,jdir=%ld) m = %Z (%ld)be_honest() failure on prime %Z Be honest for %ld primes from %ld to %ld #### Computing regulator multiple bestappr/regulator #### Tentative regulator : %Z ***** check = %f D = %Z den = %Z truncation error in bestappr #### Computing check KCZ = %ld, KC = %ld, n = %ld weighted G matricesbase change = wrong type in too_bigbnrclassbnrinitisprincipalrayplease apply bnrinit(,,1) and not bnrinit(,,0)bnfcertify Testing primes | h(K) PHASE 2: are all primes good ? Testing primes <= B (= %lu) sorry, too many primes to checklarge Minkowski bound: certification will be VERY longMinkowski bound is too largeconductorincorrect subgroup in %srnfnormgroupnon Galois extension in rnfnormgroupnot an Abelian extension in rnfnormgroupnot an Abelian extension in rnfnormgroup?rnfconductorincorrect character length in KerCharconductorofcharbnrclassnolistdiscrayabslistnot a factorisation in decodemoduleincorrect hash code in decodemoduleLbnrclassno[2]: discrayabslistarchStarting discrayabs computations Starting bnrclassno computations [1]: discrayabslistarchStarting zidealstarunits computations r1>15 in discrayabslistarchnon-positive bound in DiscrayabslistsubgrouplistBuchrayMahler bound for regulator: %Z Default bound for regulator: 0.2 (lower bound for regulator) M = %Z M0 = %Z M* = %Z pol = %Z old method: y = %Z, M0 = %Z p divides h(K) Beta list = %Z p divides w(K) *** testing p = %lu neither bnf nor bnr in conductor or discraybad subgroup in conductor or discraybnrdiscrayfactordivexact is not exact!Searching minimum of T2-form on units: BOUND = %ld [ %ld, %ld, %ld ]: %Z prime ideal Q: %Z column #%ld of the matrix log(b_j/Q): %Z new rank: %ld generator of (Zk/Q)^*: %Z zpsolubleqpsolubleqpsolublenfnfhilbertpzpsolublenf0 argument in nfhilbertp0 argument in nfhilbertnfhilbert not soluble at finite place: %Z nfhilbert not soluble at real place %ld bnfsunitbnfissunitmain variable must be of higher priority in rnfisnorminitrnfisnorminitnon monic relative equationplease apply rnfisnorminit firstrnfisnormuseless flag in rnfisnorm: the extension is Galoisconjugate %ld: %Z galoisconj2polincorrect denominator in initgaloisborne: %ZFixedField: Found: %Z p too small in fixedfieldsympolFixedField: Size: %ldx%ld GaloisConj:Fini! GaloisConj: exp %d: s=%ld [%ld] a=%ld w=%ld wg=%ld sr=%ld GaloisConj: B=%Z GaloisConj: G[%d]=%Z of relative order %d GaloisConj:Back to Earth:%Z GaloisConj:Frobenius:%Z GaloisConj:Orbite:%Z GaloisConj:Testing S4 first GaloisConj:Testing A4 first GaloisConj:denominator:%Z polynomial not in Z[X] in galoisconj4Second arg. must be integer in galoisconj4%d Calcul polynomesGaloisConj:%Z vandermondeinversemod()rootpadicfast()galoisborne()non-monic polynomial in galoisconj4galoisconj4NumberOfConjugates:card=%ld,p=%ld NumberOfConjugates:Nbtest=%ld,card=%ld,p=%ld nfgaloisconjconjugates list may be incomplete in nfgaloisconjNot a Galois field in a Galois related functionplease apply galoisinit firstfield not Galois or Galois group not weakly super solvablegaloispermtopolGaloisFixedField:den=%Z mod=%Z GaloisFixedField:cosets=%Z galoisfixedfieldpriority of optional variable too high in galoisfixedfieldGaloisConj:increase prec of p-adic roots of %ld. galoisisabelianGaloisConj: Bound %Z GaloisConj:val1=%ld val2=%ld vandermondeinversemonomorphismlift()MonomorphismLift: lift to prec %dMonomorphismLift: false early solution. MonomorphismLift: true early solution. MonomorphismLift: trying early solution %Z GaloisConj:Sortie Init Test GaloisConj:Entree Init Test Combinatorics too hard : would need %Z tests! I'll skip it but you will get a partial result...testpermutation(%Z)GaloisConj:%d hop sur %Z iterations %d%% GaloisConj:I will try %Z permutations FixedField: Sym: %Z FixedField: Weight: %Z incorrect permutation in permtopolgaloisanalysis()GaloisAnalysis:p=%ld l=%ld group=%ld deg=%ld ord=%ld GaloisAnalysis:non Galois for p=%ld Galois group almost certainly not weakly super solvableGaloisAnalysis:Nbtest=%ld,p=%ld,o=%ld,n_o=%d,best p=%ld,ord=%ld,k=%ld Polynomial not squarefree in galoisinitA4GaloisConj:%ld hop sur %d iterations max A4GaloisConj:O=%Z A4GaloisConj:orb=%Z A4GaloisConj:tau=%Z A4GaloisConj: %ld hop sur %ld iterations A4GaloisConj:sigma=%Z A4GaloisConj:I will test %ld permutations S4GaloisConj:Testing %d/8 %d:%d:%d S4GaloisConj:Testing %d/3:%d/2:%d/2:%d/4:%Z:%Z S4GaloisConj:sigma=%Z S4GaloisConj:pj=%Z S4GaloisConj:Testing %d/3:%d/4:%d/4:%d/4:%Z S4GaloisConj:Computing isomorphisms %d:%Z frobenius powerGaloisConj:next p=%ld galoisconj _may_ hang up for this polynomialGaloisConj: Fixed field %Z f=%Z borne=%Z l-borne=%Z GaloisConj: Solution too large, discard it. M%d.s4test()Best lift: %d Galoisconj:Subgroups list:%Z Trying degre %d. GaloisConj:p=%ld deg=%ld fp=%ld Combinatorics too hard : would need %Z tests! I will skip it, but it may induce an infinite loopGaloisConj: not found, %d hops GaloisConj:Testing %ZGaloisConj: %d hops on %Z tests +different pointers in ff_poltypedifferent modulus in ff_poltype/gmulsggmul2nincompatible variables in grednormalizing a series with 0 leading termmatsizegreffel <= 2 in greffegtolongcomparisonforbidden or conflicting type in gvalggvalforbidden divisor %Z in ggvalp = 1 in Z_lvalremnegationabs is not meromorphic at 0gabsempty vector in vecmaxempty vector in vecmingaffect (geuler)gaffect (gpi)gaffect (ghalf)gaffect (gi)gaffect (pol_1/pol_x)gaffect (gnil)gaffect (gen_2)gaffect (gen_m1)gaffect (gen_1)gaffect (gen_0)cvtop2not an integer modulus in cvtopcvtopgcvtopgtofpgexponormalizenormalizepolgsignenegative length in listcreatelistkilllistputno more room in this L (size %ld)negative index (%ld) in listputlistinsertbad index in listinsertno more room in this listgtolistlistconcatlistsortgvargpolvarnot the same prime in padicprecpadicprecdegreepolleadiscomplex%Mod\gdivmodinv_serinverseincorrect permtuation to inversesubst: unexpected variable precedenceforbidden substitution in a scalar typenon polynomial or series type substituted in a seriesnon positive valuation in a series substitutiongsubstforbidden substitution by a vectorforbidden substitution by a non square matrixsubstvecdifferent number of variables and values in substvecnot a variable in substvecnot a series in serreverserecipvaluation not equal to 1 in serreversederivincorrect argument in O()non-positive argument in O()incorrect object in O()zero argument in O()a log appears in intformala log/atan appears in intformalinteggfloorgceilgroundgrndtoigfloor2nisintgtruncgtosermain variable must have higher priority in gtoservectosmallnonexistent componentthis object is a leaf. It has no componentspolcoeffnon existent component in truecoeffnonexistent component in truecoeffdenomnumerliftlift_interncenterliftevaluation of a power seriesgevalsimplify_iinvalid quadratic form in qfevalinvalid vector in qfevalinvalid data in qfevalinvalid quadratic form in qfbevalinvalid vector in qfbevalinvalid data in qfbevalinvalid base change matrix in qf_base_changeinvalid data in qf_base_changeinvalid quadratic form in hqfevalinvalid vector in hqfevalinvalid data in hqfevalpolevalpoleval: i = %ldvariable must have higher priority in gtopolyt_SER with negative valuation in gtopolygtopolygreal/gimagmiller(rabin)found factor %Z currently lost to the factoring machineryMiller-Rabin: testing base %ld False prime number %Z in plisprimePL: N-1 factored! PL: proving primality of N = %Z snextpr: %lu != prc210_rp[%ld] mod 210 [caller of] snextprsnextpr: integer wraparound after prime %lu snextpr: %lu should have been prime but isn't ECM: time = %6ld ms, ellfacteur giving up. ECM: time = %6ld ms, p <= %6lu, found factor = %Z (giant step at p = %lu) (baby step table complete) (extracted precomputed helix / baby step entries) ECM: finishing curves %ld...%ld ECM: time = %6ld ms, entering B2 phase, p = %lu (got initial helix) (got [p]Q, p = %lu = prc210_rp[%ld] mod 210) ECM: %lu should have been prime but isn't ellfacteur (got [2]Q...[10]Q) ECM: time = %6ld ms, B1 phase done, p = %lu, setting up for B2 ECM: time = %6ld ms ECM: dsn = %2ld, B1 = %4lu, B2 = %6lu, gss = %4ld*420 ECM: stack tight, using heap space ECM: working on %ld curves at a time; initializing for up to %ld rounds... for one roundECM: number too small to justify this stage Rho: searching small factor of %ld-bit integer Rho: using X^2%+1ld for up to %ld rounds of 32 iterations Rho: time = %6ld ms, %3ld rounds, back to normal mode Rho: fast forward phase (%ld rounds of 64)... Rho: time = %6ld ms, Pollard-Brent giving up. sRho: time = %6ld ms, %3ld round%s Pollard-Brent failed. composite found %sfactor = %Z found factors = %Z, %Z, and %Z Rho: hang on a second, we got something here... found factor = %Z Rho: restarting for remaining rounds... SQUFOF: giving up, time = %ld ms SQUFOF: second cycle exhausted after %ld iterations, dropping it SQUFOF: found factor %ld^2 SQUFOF: ...but the root form seems to be on the principal cycle SQUFOF: square form (%ld^2, %ld, %ld) on second cycle after %ld iterations, time = %ld ms SQUFOF: first cycle exhausted after %ld iterations, dropping it SQUFOF: square form (%ld^2, %ld, %ld) on first cycle after %ld iterations, time = %ld ms SQUFOF: blacklisting a = %ld on second cycle SQUFOF: blacklisting a = %ld on first cycle SQUFOF: entering main loop with forms (1, %ld, %ld) and (1, %ld, %ld) of discriminants %Z and %Z, respectively squfof [caller of] (5n is a square)squfof [caller of] (n or 3n is a square) But it nevertheless wasn't a %ld%s power. %3ld: %3ld (3rd %ld, 5th %ld, 7th %ld) OddPwrs: is %Z ...a, or 3rd%s, or 5th%s power? modulo: resid. (remaining possibilities) 7th But it wasn't a pure power. OddPwrs: passed modular checks - ruled out checking modulo %ld OddPwrs: testing for exponent %ld OddPwrs: examining %Z ifac_startfactoring 0 in ifac_startIFAC: new partial factorization structure (%ld slots) ... factor no. %ld is a duplicate%s (so far)... factor no. %ld was unique%s stored (largest) factor no. %ld... IFAC: incorporating set of %ld factor(s) ifac_decompIFAC: found %ld large prime (power) factor%s. [2] ifac_decompIFAC: (Partial fact.)Stop requested. factoring 0 in ifac_decompifac_moebiusifac_issquarefreeifac_omegaifac_bigomegafactor has NULL exponent in ifac_findifac_totientifac_numdivifac_sumdivifac_sumdivk[caller of] elladd0SQUFOF: found factor %ld from ambiguous form after %ld steps on the ambiguous cycle, time = %ld ms SQUFOF: squfof_ambig returned %ld SQUFOF: ...found nothing on the ambiguous cycle after %ld steps there, time = %ld ms IFAC: main loop: repeated old factor %Z non-existent factor class in ifac_mainIFAC: main loop: another factor was divisible by %Z IFAC: main loop: repeated new factor %Z IFAC: after main loop: repeated old factor %Z IFAC: main loop: %ld factor%s left IFAC: main loop: this was the last factor LucasModIFAC: found %Z = %Z ^2 IFAC: found %Z = %Z ^%ld ifac_crack [Z_issquarerem miss]IFAC: cofactor = %Z IFAC: factoring %Z yielded `factor' %Z which isn't! factoringIFAC: forcing ECM, may take some time IFAC: unfactored composite declared prime %Z IFAC: untested integer declared primeIFAC: trying MPQS IFAC: trying Lenstra-Montgomery ECM IFAC: trying Shanks' SQUFOF, will fail silently if input is too large for it. IFAC: trying Pollard-Brent rho method IFAC: factor %Z is prime IFAC: checking for odd power IFAC: checking for pure square IFAC: cracking composite %Z IFAC: prime %Z appears at least to the power %ld IFAC: a factor was divisible by another prime factor, leaving a cofactor = %Z IFAC: a factor was a power of another prime factor IFAC: prime %Z appears with exponent = %ld ifac_sort_one`*where' out of bounds in ifac_sort_one`washere' out of bounds in ifac_sort_onemisaligned partial detected in ifac_sort_oneprime equals composite in ifac_sort_onecomposite equals prime in ifac_sort_oneIFAC: repeated factor %Z detected in ifac_sort_one partial impossibly short in ifac_sort_oneIFAC: factor %Z is prime (no larger composite) compositeprimeIFAC: factor %Z is %s avoiding nonexistent factors in ifac_whoiswhovecsmall_copyperm_to_GAP%ldempty group in group_domaincoset not found in cosets_perm_searchgaloissubgroup: not a WSS groupwrong argument in galoisisabelianGroup(, )Group(())PermutationGroup<|>PermutationGroup<1|>galoisexporteuclidean division (poldivrem)factmodnot a prime in rootmodprime too big in rootmod2not a prime in factmodnot a prime in FpX_quad_rootpolrootsmodBerlekamp_kerBerlekamp_matrixFpXQYQ_powkernelBerlekamp matrix %3ld factor of degree %3ld %3ld fact. of degree %3ld frobeniusspec_FpXQ_powzero polynomial in FpXQ_pow. %Z not primelx= %ldsubresall, dr = %ldsubresallsubresext, dr = %ldinexact computation in subresextsubresextRgX_extgcd, dr = %ldinexact computation in RgX_extgcdresultantducos, degpol Q = %ldsylvestermatrixnot the same variables in sylvestermatrixpolresultantsrgcdsrgcd: dr = %ld discsrreduceddiscsmithpoldiscreducednon-monic polynomial in poldiscreducedsturmpolsturm, dr = %ldnot a squarefree polynomial in sturmginvmodnewtonpolymatratliftpolratliftnfgcdnfgcd: p=%d lifting to prec %ldbuilding treeMultiLift: bad args ### K = %d, %Z combinations T found factor %Z remaining modular factor(s): %ld LLL_cmbfLLL_cmbf: chk_factors failed* Time LLL: %ld * Time Check Factor: %ld LLL_cmbf: rank decrease for this block of tracesLLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld LLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld) DDF: wrong numbers of factorsTime setup: %ld Total Time: %ld =========== splitting mod p = %ldfactors...tried prime %3ld (%-3ld %s). Time = %ld pseudorem dx = %ld >= %ldnextSousResultant j = %ld/%ldnon-invertible polynomial in RgXQ_invrelatively prime polynomials expectedLLL_cmbf: checking factor %ld special_pivot output: %Z Rg_to_FpRg_to_FlRg_to_FpXQnon invertible polynomial in FpXQ_invFpX_FpXQV_compo: %d FpXQ_mul [%d] powers is only [] or [1] in FpX_FpXQV_compoFpXQX_gcdnon-invertible polynomial in FpXQX_gcdFpXQ_sqrtlFpXQ_sqrtnfflgen1/0 exponent in FpXQ_sqrtnbad degrees in FpX_ffintersect: %d,%d,%dPolynomials not irreducible in FpX_ffintersectpows [P,Q]ZZ_%Z[%Z]/(%Z) is not a field in FpX_ffintersectfactor_irredfactor_irred_matFpXQ_matrix_powsFpX_resultant (da = %ld)polint_triv2 (i = %ld)FpV_polintresultant mod %ld (bound 2^%ld, stable=%ld)ZY_ZXY_rnfequationFinal lambda = %ld Degree list for ERS (trials: %ld) = %Z bound for resultant coeffs: 2^%ld Trying lambda = %ld ZY_ZXY_resultant_all: LERS needs lambdaresultant mod %ld (bound 2^%ld, stable = %d)ZX_resultantbound for resultant: 2^%ld modulargcddifferent variables in modulargcdgcd mod %lu (bound 2^%ld)bound 2^%ld. Goal 2^%ldmodulargcd: trial division failedQXQ_invQXQ_inv: mod %ld (bound 2^%ld)QXQ_inv: char 0 check failedffinitFFInit: using subcyclo(%ld, %ld) non positive degree in ffinitFF l-Gen:next %Z matrix cyclopol[Frobenius]FFTinitFFTcauchy_boundisrealapprroots (conjugates)invalid coefficients in rootspolrootsall_roots: restarting, i = %ld, e = %ld conformal_polparametersrefine_Frefine_HSubCyclo: conductor:%ld SubCyclo: new conductor:%ld SubCyclo: %ld not found SubCyclo: testing %ld^%ld padicsqrtnlift.Subcyclo: val=%ld Subcyclo: borne=%Z Subcyclo: prime l=%Z bnr must be over Q in bnr_to_znstargaloissubcycloPlease do not try to break PARI with ridiculous counterfeit data. Thanks!wrong type in galoissubcycloSubcyclo: %ld orbits with %ld elements each Subcyclo: orbits=%Z znstar_cosetsznstar_conductorSubcyclo: complex=%ld Subcyclo: elements:generators must be prime to conductor in galoissubcyclowrong modulus in galoissubcyclonot a HNF matrix in galoissubcycloMatrix of wrong dimensions in galoissubcycloN must be a bnrinit or a znstar if H is a matrix in galoissubcyclodegree <= 0 in galoissubcyclonon-cyclic case in polsubcyclo: use galoissubcyclo insteadroots_to_polsubcyclo_cyclicsubcyclo_rootsdegree does not divide phi(n) in subcyclosubcycloforsubgroupnot a group in forsubgroupnb subgroup = %ld (lifted) subgp of prime to %Z part: infinite group in forsubgroup alpha_lambda(mu,p) = %Z countsub = %ld for this type alpha = %Z forsubgroup (alpha != countsub) lambda = %ld lambda'= mu = mu'= group:subgroupsubgroup: index bound must be positiveexact type in subgrouplist subgroup: column selection:a transcendental functionincorrect precision in transcgpowgpow: underflow or overflowgpow: 0 to a non positive exponentgpow: 0 to a forbidden powergpow: nth-root does not existgpow: modulus %Z is not primegpow: need integer exponent if series valuation != 0%Z should divide valuation (= %ld) in sqrtnvaluation overflow in sqrtnsqrtrodd exponent in p-adic sqrtsqrtnrsecond arg must be integer in gsqrtnnth-root does not exist in gsqrtngsqrtn1/0 exponent in gsqrtngexpp-adic argument out of range in gexpagm of two vector/matricesnon positive argument in mplognot a p-adic argument in teichmullerzero argument in palogloglog is not meromorphic at 0zero argument in mploggloggcosp-adic argument out of range in gcosgsinp-adic argument out of range in gsinnon zero exponent in gsincosgsincoscan't compute tan(Pi/2 + kPi)gtangcotancotan0 argument in cotandegree overflow in pow_monomeser_powpadic_sqrtmpsc1gatangasingacosrfix (conversion to t_REAL)gargzero argument in garggchgthgashgachgathsingular argument in atanhBernoullicaching Bernoulli numbers 2*%ld to 2*%ld, prec = %ld argument too large in ggamdggamdgamd of a power seriesargument too large in ggammanon-positive integer argument in ggammaggammaGamma not defined for non-integral p-adic numbernon-positive integer in glngammaglngammalngamma around a!=1p-adic lngamma functionBernoulli sumsum from 0 to N-1lim, nn: [%ld, %ld] non-positive integer argument in cxpsigpsipsi of power seriesbernvec for n > 46340gammaBernoullisproduct from 0 to N-1non-positive integer argument in cxgammalim, nn: [%ld, %ld], la = %lf p-adic jbessel functionjbesselnot an integer index in jbesselhp-adic jbesselh functionjbesselhzero argument in a k/n bessel functionp-adic kbessel functioncannot give a power series result in k/n bessel functionkbesselbesselkhyperu's third argument must be positiveincgam2incgamcnon-real argument in eint1nstop = %ld negative or zero constant in veceint1Entering veceint1: erfcinv_szeta_euler, p = %lu/%luczetasum from 1 to N-1tab[q^-s] from 1 to N-1twistpartialzeta (3), j = %ld/%ldtwistpartialzeta (2), j = %ld/%ldtwistpartialzeta (1), j = %ld/%ldgzetazeta of power seriesargument equal to one in zetapolylognegative index in polylogpadic polylogarithmpolylog around a!=0gpolyloggpolylogzargument must belong to upper half-planebad argument for modular functiontrueetaweberq >= 1 in thetak < 0 in thetanullkjbessel around a!=0kbessel around a!=0szetaetanon-positive valuation in etaid too long in a stringified flaga stringified flag does not start with an idnumeric id in a stringified flagSingleton id non-single in a stringified flagNon-numeric argument of an action in a templateerror in parse_option_stringJunk after an id in a stringified flagUnrecognized action in a templateCannot negate id=value in a stringified flagUnrecognized id '%s' in a stringified flagbreak not allowedunused characters: %sunused characters[install] identifier '%s' already in use[install] updating '%s' prototype; module not reloadednot a valid identifiercan't kill thatseqcan't allow allocatemem() in loopsnot an integershift operand too bigexprexpected character: '%c' instead ofthis should be an integerinteger too bigbreak not allowed after ^not a proper member definitionbreak not allowed after !break not allowed after #I can't remember before the big bangincorrect vector or matrixbreak not allowed in array contexttruc(): n = %ldbreak not allowed in assignmentcan't derive thisbreak not allowed here (reading arguments)break not allowed here (reading long)not enough flags in string function signaturenot a variable:identifier (unknown code)Obreak not allowed in O()ifbreak not allowed in test expressionwhileuntilglobalbreak not allowed here (defining global var)symbol already in uselocalunknown function '%s', expected '=' instead oflocal(local() bloc must appear before any other expressionthis function uses a killed variabletoo many parameters in user-defined function callbreak not allowed here (reading function args)no more variables availablecan't pop gp variablepanic%s already exists with incompatible valencevariable number too bigrenaming a GP variable is forbidden; or ] expectedglobal variable not allowedskipidentifier (unknown code)using obsolete function %sa1a2a3a4a6areab2b4b6b8bidbnfc4c6clgpcodiffcycdiffdisceffufutugengroupindexjmodnfnoomegaordersppolr1r2regsignt2tatetutufuwzkzkstcan't modify a pre-defined member: unknown member functionincorrect type in %spositive integer expectedcan't replace an existing symbol by an aliasonly functions can be aliasedunknown function#) = ); .%s = this code has to come firstunknown parser codebreak not allowed here (expanding string)unfinished stringbreak not allowed in print()array index (%ld) out of allowed range %s[1-%ld][1][none]a 0x0 matrix has no elementsincorrect type or length in matrix assignmentnot a suitable VECSMALL componentglobal variable: user function %s: variable %Z declared twiceformal derivationO(a^b)=o(a^b)=p-adic or power series zero with precision given by babsGpabs(x)=absolute value (or modulus) of xacosacos(x)=inverse cosine of xacoshacosh(x)=inverse hyperbolic cosine of xaddellGGGaddell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve eaddprimesGaddprimes(x)=add primes in the vector x (with at most 20 components) to the prime tableadjadj(x)=adjoint matrix of xagmGGpagm(x,y)=arithmetic-geometric mean of x and yakellGGakell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve ealgdepGLpalgdep(x,n)=algebraic relations up to degree n of xalgdep2GLLpalgdep2(x,n,dec)=algebraic relations up to degree n of x where dec is as in lindep2algtobasisalgtobasis(nf,x)=transforms the algebraic number x into a column vector on the integral basis nf[7]anellGLanell(e,n)=computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<32768)apellapell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's methodapell2apell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbolsapprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod pargarg(x)=argument of x,such that -pi0 in the wide sense. See manual for the other parameters (which can be omitted)bytesizebytesize(x)=number of bytes occupied by the complete tree of the object xceilceil(x)=ceiling of x=smallest integer>=xcenterlift(x)=centered lift of x. Same as lift except for integermodscfcf(x)=continued fraction expansion of x (x rational,real or rational function)cf2cf2(b,x)=continued fraction expansion of x (x rational,real or rational function), where b is the vector of numerators of the continued fractionchangevarchangevar(x,y)=change variables of x according to the vector ycharGnchar(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the comatrixchar1char1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange interpolationchar2char2(x,y)=characteristic polynomial of the matrix x expressed with variable y, using the Hessenberg form. Can be much faster or much slower than char, depending on the base ringchellchell(x,y)=change data on elliptic curve according to y=[u,r,s,t]chinese(x,y)=x,y being integers modulo mx and my,finds z such that z is congruent to x mod mx and y mod mychptellchptell(x,y)=change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]classno(x)=class number of discriminant xclassno2(x)=class number of discriminant xcoeffcoeff(x,s)=coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[])compimagcompimag(x,y)=Gaussian composition of the binary quadratic forms x and y of negative discriminantcompocompo(x,s)=the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]compositum(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2compositum2compositum2(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2, with roots of pol1 and pol2 expressed on the compositum polynomialscomprealrawcomprealraw(x,y)=Gaussian composition without reduction of the binary quadratic forms x and y of positive discriminantconcat(x,y)=concatenation of x and yGDGDGD1,G,conductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupconductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrconjconj(x)=the algebraic conjugate of xconjvec(x)=conjugate vector of the algebraic number xcontent(x)=gcd of all the components of x, when this makes senseconvolconvol(x,y)=convolution (or Hadamard product) of two power seriescorecore(n)=unique (positive of negative) squarefree integer d dividing n such that n/d is a squarecore2core2(n)=(long)gen_2-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisccoredisc(n)=discriminant of the quadratic field Q(sqrt(n))coredisc2coredisc2(n)=(long)gen_2-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercoscos(x)=cosine of xcoshcosh(x)=hyperbolic cosine of xcvtoicvtoi(x)=truncation of x, without taking into account loss of integer part precisioncycloLDncyclo(n)=n-th cyclotomic polynomialdecodefactordecodefactor(fa)=given a factorisation fa, gives the factored object backdecodemoduledecodemodule(nf,fa)=given a coded module fa as in discrayabslist, gives the true moduledegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardenom(x)=denominator of x (or lowest common denominator in case of an array)deplin(x)=finds a linear dependence between the columns of the matrix xderiv(x,y)=derivative of x with respect to the main variable of ydet(x)=determinant of the matrix xdet2(x)=determinant of the matrix x (better for integer entries)detint(x)=some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with hermitemoddiagonaldiagonal(x)=creates the diagonal matrix whose diagonal entries are the entries of the vector xdilogdilog(x)=dilogarithm of xdirdivdirdiv(x,y)=division of the Dirichlet series x by the Dir. series ydireulerV=GGIDGdireuler(p=a,b,expr)=Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s)dirmul(x,y)=multiplication of the Dirichlet series x by the Dir. series ydirzetakdirzetak(nf,b)=Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1disc(x)=discriminant of the polynomial xdiscfdiscf(x)=discriminant of the number field defined by the polynomial x using round 4discf2discf2(x)=discriminant of the number field defined by the polynomial x using round 2discrayabsGD0,G,D0,G,D0,L,discrayabs(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayabscondGD0,G,D0,G,D2,L,discrayabscond(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if fmodule is not the conductordiscrayabslist(bnf,listes)=if listes is a 2-component vector as output by ideallistunit or similar, gives list of corresponding discrayabsconddiscrayabslistarchGGLdiscrayabslistarch(bnf,arch,bound)=gives list of discrayabscond of all modules up to norm bound with archimedean places arch, in a longvector formatdiscrayabslistarchalldiscrayabslistarchall(bnf,bound)=gives list of discrayabscond of all modules up to norm bound with all possible archimedean places arch in reverse lexicographic order, in a longvector formatdiscrayabslistlongdiscrayabslistlong(bnf,bound)=gives list of discrayabscond of all modules up to norm bound without archimedean places, in a longvector formatdiscrayrelGD0,G,D0,G,D1,L,discrayrel(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayrelcondGD0,G,D0,G,D3,L,discrayrelcond(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if module is not the conductordivisors(x)=gives a vector formed by the divisors of x in increasing orderdivresdivres(x,y)=euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainderdivsumGVIdivsum(n,X,expr)=sum of expression expr, X running over the divisors of neigen(x)=eigenvectors of the matrix x given as columns of a matrixeint1eint1(x)=exponential integral E1(x)erfc(x)=complementary error functioneta(x)=eta function without the q^(1/24)eulereuler=euler()=euler's constant with current precisionevaleval(x)=evaluation of x, replacing variables by their valueexpexp(x)=exponential of xextract(x,y)=extraction of the components of the vector x according to the vector or mask y, from left to right (1, 2, 4, 8, ...for the first, second, third, fourth,...component)factfact(x)=factorial of x (x C-integer), the result being given as a real numberfactcantorfactcantor(x,p)=factorization mod p of the polynomial x using Cantor-Zassenhausfactfqfactfq(x,p,a)=factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factmod(x,p)=factorization mod p of the polynomial x using Berlekampfactorfactor(x)=factorization of xfactoredbasisGGffactoredbasis(x,p)=integral basis of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoreddiscffactoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoredpolredfactoredpolred(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives minimal polynomials only)factoredpolred2factoredpolred2(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives elements and minimal polynomials)factornffactornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial tfactorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r, using the round 4 algorithmfactorpadic2factorpadic2(x,p,r)=p-adic factorization of the polynomial x to precision r, using Buchmann-LenstraGLLfactpol(x,l,hint)=factorization over Z of the polynomial x up to degree l (complete if l=0) using Hensel lift, knowing that the degree of each factor is a multiple of hintfactpol2factpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0) using root findingfibofibo(x)=fibonacci number of index x (x C-integer)floorfloor(x)=floor of x=largest integer<=xforvV=GGIfor(X=a,b,seq)=the sequence is evaluated, X going from a up to bfordivvGVIfordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of nforprimeforprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and bforstepvV=GGGIforstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of sforvecvV=GID0,L,forvec(x=v,seq)=v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1fpnGLDnfpn(p,n)=monic irreducible polynomial of degree n over F_p[x]fracfrac(x)=fractional part of x=x-floor(x)galois(x)=Galois group of the polynomial x (see manual for group coding)galoisapply(nf,aut,x)=Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfgaloisconjgaloisconj(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field, using p-adics, LLL on integral basis (not always complete)galoisconj1galoisconj1(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field nf, using complex numbers, LLL on integral basis (not always complete)galoisconjforcegaloisconjforce(nf)=list of conjugates of a root of the polynomial x=nf[1] in the Galois number field nf, using p-adics, LLL on integral basis. Guaranteed to be complete if the field is Galois, otherwise there is an infinite loopgamhgamh(x)=gamma of x+1/2 (x integer)gamma(x)=gamma function at xgauss(a,b)=gaussian solution of ax=b (a matrix,b vector)gaussmodulo(M,D,Y)=(long)gen_1 solution of system of congruences MX=Y mod Dgaussmodulo2gaussmodulo2(M,D,Y)=all solutions of system of congruences MX=Y mod Dgcd(x,y)=greatest common divisor of x and ygetheapgetheap()=2-component vector giving the current number of objects in the heap and the space they occupygetrandlgetrand()=current value of random number seedgetstackgetstack()=current value of stack pointer avmagettimegettime()=time (in milliseconds) since last call to gettimeglobalredglobalred(e)=e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pgotos*goto(n)=THIS FUNCTION HAS BEEN SUPPRESSEDhclassno(x)=Hurwitz-Kronecker class number of x>0hellhell(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using theta-functionshell2hell2(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using Tate's methodhermitehermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, using a naive algorithmhermite2hermite2(x)=2-component vector [H,U] such that H is an (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, and U is a unimodular matrix such that xU=H, using Batut's algorithmhermitehavashermitehavas(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Havas's algorithmhermitemodhermitemod(x,d)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is the non-zero determinant of this latticehermitemodidhermitemodid(x,d)=(upper triangular) Hermite normal form of x concatenated with d times the identity matrixhermitepermhermiteperm(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Batut's algorithmhess(x)=Hessenberg form of xhilblGGGhilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)hilberthilbert(n)=Hilbert matrix of order n (n C-integer)hilbplGGhilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)hvectorhvector(n,X,expr)=row vector with n components of expression expr, the variable X ranging from 1 to nhyperuhyperu(a,b,x)=U-confluent hypergeometric functionii=i()=square root of -1idealaddidealadd(nf,x,y)=sum of two ideals x and y in the number field defined by nfidealaddmultoneidealaddone(nf,x,y)=when the sum of two ideals x and y in the number field K defined by nf is equal to Z_K, gives a two-component vector [a,b] such that a is in x, b is in y and a+b=1idealaddoneidealaddmultone(nf,list)=when the sum of the ideals in the number field K defined by nf and given in the vector list is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1idealappridealappr(nf,x)=x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealapprfactidealapprfact(nf,x)=x being a prime ideal factorization with possibly zero or negative exponents, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealchineseidealchinese(nf,x,y)=x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprimeidealcoprime(nf,x,y)=gives an element b in nf such that b.x is an integral ideal coprime to the integral ideal yidealdiv(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nfidealdivexactidealdivexact(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nf when the quotient is known to be an integral idealidealfactoridealfactor(nf,x)=factorization of the ideal x given in HNF into prime ideals in the number field nfidealhermite(nf,x)=hermite normal form of the ideal x in the number field nf, whatever form x may haveidealhermite2idealhermite2(nf,a,b)=hermite normal form of the ideal aZ_K+bZ_K in the number field K defined by nf, where a and b are elementsidealintersectidealintersect(nf,x,y)=intersection of two ideals x and y in HNF in the number field defined by nfidealinv(nf,x)=inverse of the ideal x in the number field nf not using the differentidealinv2idealinv2(nf,x)=inverse of the ideal x in the number field nf using the differentideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfideallistarch(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, without generatorsideallistarchgenideallistarchgen(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, with generatorsideallistunitideallistunit(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, without generatorsideallistunitarchideallistunitarch(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitideallistunitarchgenideallistunitarchgen(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitgenideallistunitgenideallistunitgen(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, with generatorsideallistzstarideallistzstar(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, without generatorsideallistzstargenideallistzstargen(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, with generatorsideallllredideallllred(nf,x,vdir)=LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealmulidealmul(nf,x,y)=product of the two ideals x and y in the number field nfidealmulredidealmulred(nf,x,y)=reduced product of the two ideals x and y in the number field nfidealnorm(nf,x)=norm of the ideal x in the number field nfidealpowidealpow(nf,x,n)=n-th power of the ideal x in HNF in the number field nfidealpowredidealpowred(nf,x,n)=reduced n-th power of the ideal x in HNF in the number field nfidealtwoeltidealtwoelt(nf,x)=(long)gen_2-element representation of an ideal x in the number field nfidealtwoelt2idealtwoelt2(nf,x,a)=(long)gen_2-element representation of an ideal x in the number field nf, with the first element equal to aidealval(nf,x,p)=valuation at p given in primedec format of the ideal x in the number field nfidmatidmat(n)=identity matrix of order n (n C-integer)if(a,seq1,seq2)=if a is nonzero, seq1 is evaluated, otherwise seq2imagimag(x)=imaginary part of ximageimage(x)=basis of the image of the matrix ximage2(x)=basis of the image of the matrix ximagecomplimagecompl(x)=vector of column indices not corresponding to the indices given by the function imageincgamincgam(s,x)=incomplete gamma functionincgam1incgam1(s,x)=incomplete gamma function (for debugging only)incgam2(s,x)=incomplete gamma function (for debugging only)incgam3incgam3(s,x)=complementary incomplete gamma functionincgam4incgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputedindexrankindexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such that the exracted matrix is square of maximal rankindsortindsort(x)=indirect sorting of the vector xinitalginitalg(x)=x being a nonconstant irreducible polynomial, gives the vector: [x,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgredinitalgred(x)=x being a nonconstant irreducible polynomial, finds (using polred) a simpler polynomial pol defining the same number field, and gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual), r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgred2initalgred2(P)=P being a nonconstant irreducible polynomial, gives a two-element vector [nf,mod(a,pol)], where nf is as output by initalgred and mod(a,pol) is a polymod equal to mod(x,P) and pol=nf[1]initellinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]initzetainitzeta(x)=compute number field information necessary to use zetak, where x is an irreducible polynomialinteg(x,y)=formal integration of x with respect to the main variable of yintersect(x,y)=intersection of the vector spaces whose bases are the columns of x and yintgenV=GGID1,L,pintgen(X=a,b,s)=general numerical integration of s from a to b with respect to X, to be used after removing singularitiesintinfV=GGID2,L,pintinf(X=a,b,s)=numerical integration of s from a to b with respect to X, where a or b can be plus or minus infinity (1.0e4000), but of same signintnumV=GGID0,L,pintnum(X=a,b,s)=numerical integration of s from a to b with respect to XintopenV=GGID3,L,pintopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or binverseimage(x,y)=an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwiseisdiagonal(x)=true(1) if x is a diagonal matrix, false(0) otherwiseisfundisfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if notisidealisideal(nf,x)=true(1) if x is an ideal in the number field nf, false(0) if notisinclisincl(x,y)=tests whether the number field defined by the polynomial x is isomorphic to a subfield of the one defined by y; 0 if not, otherwise all the isomorphismsisinclfastisinclfast(nf1,nf2)=tests whether the number nf1 is isomorphic to a subfield of nf2 or not. If it gives a non-zero result, this proves that this is the case. However if it gives zero, nf1 may still be isomorphic to a subfield of nf2 so you have to use the much slower isincl to be sureisirreducibleisirreducible(x)=true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantisisomisisom(x,y)=tests whether the number field defined by the polynomial x is isomorphic to the one defined by y; 0 if not, otherwise all the isomorphismsisisomfastisisomfast(nf1,nf2)=tests whether the number fields nf1 and nf2 are isomorphic or not. If it gives a non-zero result, this proves that they are isomorphic. However if it gives zero, nf1 and nf2 may still be isomorphic so you have to use the much slower isisom to be sureisoncurveiGGisoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if notisprimeGD0,L,isprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if notisprincipal(bnf,x)=bnf being output by buchinit, gives the vector of exponents on the class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalforceisprincipalforce(bnf,x)=same as isprincipal, except that the precision is doubled until the result is obtainedisprincipalgenisprincipalgen(bnf,x)=bnf being output by buchinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorisprincipalgenforceisprincipalgenforce(bnf,x)=same as isprincipalgen, except that the precision is doubled until the result is obtainedisprincipalray(bnf,x)=bnf being output by buchrayinit, gives the vector of exponents on the ray class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalraygenisprincipalraygen(bnf,x)=bnf being output by buchrayinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorispspispsp(x)=true(1) if x is a strong pseudoprime, false(0) if notisqrtisqrt(x)=integer square root of x (x integer)issetisset(x)=true(1) if x is a set (row vector with strictly increasing entries), false(0) if notissqfreeissqfree(x)=true(1) if x is squarefree, false(0) if notissquareissquare(x)=true(1) if x is a square, false(0) if notisunit(bnf,x)=bnf being output by buchinit, gives the vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisejacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xjbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non-negative integerjelljell(x)=elliptic j invariant of xkaramulkaramul(x,y,k)=THIS FUNCTION HAS BEEN SUPPRESSEDkbessel(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)kbessel2kbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)kerker(x)=basis of the kernel of the matrix xkeri(x)=basis of the kernel of the matrix x with integer entrieskerintkerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkerint1kerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational entries using matrixqz3 and the HNFkerint2kerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkrokro(x,y)=kronecker symbol (x/y)labellabel(n)=THIS FUNCTION HAS BEEN SUPPRESSEDlambdaklambdak(nfz,s)=Dedekind lambda function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)laplacelaplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^nlcmlcm(x,y)=least common multiple of x and y=x*y/gcd(x,y)legendre(n)=legendre polynomial of degree n (n C-integer)lengthlength(x)=number of non code words in xlexlex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwisematrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xmatrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsize(x)=number of rows and columns of the vector/matrix x as a 2-vectormaxmax(x,y)=maximum of x and yminmin(x,y)=minimum of x and yminidealminideal(nf,ix,vdir)=minimum of the ideal ix in the direction vdir in the number field nfminimminim(x,bound,maxnum)=number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0minim2minim2(x,bound)=looks for vectors of square norm <= bound, return the first one and its normmod(x,y)=creates the integer x modulo y on the PARI stackmodpmodp(x,y)=creates the integer x modulo y as a permanent object (on the heap)modreversemodreverse(x)=reverse polymod of the polymod x, if it existsmodulargcd(x,y)=gcd of the polynomials x and y using the modular methodmumu(x)=Moebius function of xnewtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnextprimenextprime(x)=smallest prime number>=xnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfdiv(nf,a,b)=element a/b in nfnfdiveucnfdiveuc(nf,a,b)=gives algebraic integer q such that a-bq is smallnfdivresnfdivres(nf,a,b)=gives [q,r] such that r=a-bq is smallnfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhermitemod(nf,x,detx)=if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfmodnfmod(nf,a,b)=gives r such that r=a-bq is small with q algebraic integernfmul(nf,a,b)=element a.b in nfnfpownfpow(nf,a,k)=element a^k in nfnfreducenfreduce(nf,a,id)=gives r such that a-r is the ideal id and r is smallnfsmith(nf,x)=if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfvalnfval(nf,a,pr)=valuation of element a at the prime prnormnorm(x)=norm of xnorml2norml2(x)=square of the L2-norm of the vector xnucompnucomp(x,y,l)=composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputednumdivnumdiv(x)=number of divisors of xnumer(x)=numerator of xnupownupow(x,n)=n-th power of primitive positive definite quadratic form x using nucomp and nuduploo(a^b)=O(a^b)=p-adic or power series zero with precision given by bomega(x)=number of unrepeated prime divisors of xordellordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve eorderorder(x)=order of the integermod x in (Z/nZ)*orderellorderell(e,p)=order of the point p on the elliptic curve e over Q, 0 if non-torsionordred(x)=reduction of the polynomial x, staying in the same orderpadicprec(x,p)=absolute p-adic precision of object xpascalLDGpascal(n)=pascal triangle of order n (n C-integer)perfperf(a)=rank of matrix of xx~ for x minimal vectors of a gram matrix apermutationLGpermutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)permutation2numpermutation2num(vect)=ordinal (between 1 and n!) of permutation vectpfpf(x,p)=returns the prime form whose first coefficient is p, of discriminant xphiphi(x)=Euler's totient function of xpipi=pi()=the constant pi, with current precisionpnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction xpointellpointell(e,z)=coordinates of point on the curve e corresponding to the complex number zpolintGGGD&polint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, yapolred(x)=reduction of the polynomial x (gives minimal polynomials only)polred2polred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)polredabspolredabs(x)=a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 normpolredabs2polredabs2(x)=gives [pol,a] where pol is as in polredabs, and alpha is the element whose characteristic polynomial is polpolredabsallpolredabsall(x)=complete list of the smallest generating polynomials of the number field for the T2 norm on the rootspolredabsfastpolredabsfast(x)=a smallest generating polynomial of the number field for the T2 norm on the rootspolredabsnoredpolredabsnored(x)=a smallest generating polynomial of the number field for the T2 norm on the roots without initial polredpolsym(x,n)=vector of symmetric powers of the roots of x up to npolvarpolvar(x)=main variable of object x. Gives p for p-adic x, error for scalarspolypoly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientLGppolylog(m,x)=m-th polylogarithm of xpolylogdpolylogd(m,x)=D_m~-modified m-th polylog of xpolylogdoldpolylogdold(m,x)=D_m-modified m-th polylog of xpolylogppolylogp(m,x)=P_m-modified m-th polylog of xpolyrevpolyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termpolzagLLpolzag(n,m)=Zagier's polynomials of index n,mpowellpowell(e,x,n)=n times the point x on elliptic curve e (n in Z)powrealrawpowrealraw(x,n)=n-th power without reduction of the binary quadratic form x of positive discriminantprecprec(x,n)=change the precision of x to be n (n C-integer)precisionprecision(x)=real precision of object xprime(n)=returns the n-th prime (n C-integer)primedec(nf,p)=prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a.Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantprimesprimes(n)=returns the vector of the first n primes (n C-integer)primrootprimroot(n)=returns a primitive root of n when it existsprincipalideal(nf,x)=returns the principal ideal generated by the algebraic number x in the number field nfprincipalideleprincipalidele(nf,x)=returns the principal idele generated by the algebraic number x in the number field nfprodGV=GGIprod(x,X=a,b,expr)=x times the product (X runs from a to b) of expressionprodeulerV=GGIpprodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expressionprodinfV=GID0,L,pprodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expressionprodinf1V=GID1,L,pprodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expressionpsipsi(x)=psi-function at xqfiqfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0qfrGGGGqfr(a,b,c,d)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c>0 and distance dquaddiscquaddisc(x)=discriminant of the quadratic field Q(sqrt(x))quadgenquadgen(x)=standard generator of quadratic order of discriminant xquadpoly(x)=quadratic polynomial corresponding to the discriminant xDGrandom()=random integer between 0 and 2^31-1rankrank(x)=rank of the matrix xrayclassnorayclassno(bnf,x)=ray class number of the module x for the big number field bnf. Faster than buchray if only the ray class number is wantedrayclassnolistrayclassnolist(bnf,liste)=if listes is as output by idealisunit or similar, gives list of corresponding ray class numbersrealreal(x)=real part of xrecip(x)=reciprocal polynomial of xredimag(x)=reduction of the binary quadratic form x with D<0redrealredreal(x)=reduction of the binary quadratic form x with D>0redrealnodredrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]reduceddiscreduceddisc(f)=vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fregula(x)=regulator of the real quadratic field of discriminant xreorderreorder(x)=reorder the variables for output according to the vector xresultantresultant(x,y)=resultant of the polynomials x and y with exact entriesresultant2resultant2(x,y)=resultant of the polynomials x and yreversereverse(x)=reversion of the power series xrhorealrhoreal(x)=single reduction step of the binary quadratic form x of positive discriminantrhorealnodrhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]rndtoirndtoi(x)=take the nearest integer to all the coefficients of x, without taking into account loss of integer part precisionrnfbasis(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfdiscfrnfdiscf(nf,pol)=given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfequation(nf,pol)=given a pol with coefficients in nf, gives the absolute equation of the number field defined by polrnfequation2rnfequation2(nf,pol)=given a pol with coefficients in nf, gives [apol,th], where apol is the absolute equation of the number field defined by pol and th expresses the root of nf[1] in terms of the root of apolrnfhermitebasisrnfhermitebasis(bnf,order)=given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfisfree(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnflllgram(nf,pol,order)=given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfpolred(nf,pol)=given a pol with coefficients in nf, finds a list of polynomials defining some subfields, hopefully simplerrnfpseudobasisrnfpseudobasis(nf,pol)=given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitz(nf,order)=given an order as output by rnfpseudobasis, gives [A,I,..] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialrootmodrootmod(x,p)=roots mod p of the polynomial xrootmod2rootmod2(x,p)=roots mod p of the polynomial x, when p is smallrootpadic(x,p,r)=p-adic roots of the polynomial x to precision rroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonrootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfrootsold(x)=roots of the polynomial x using a modified Newton's methodroundround(x)=take the nearest integer to all the coefficients of xrounderrorrounderror(x)=maximum error found in rounding xseriesseries(x,v)=convert x (usually a vector) into a power series with variable v, starting with the constant coefficientsetset(x)=convert x into a set, i.e. a row vector with strictly increasing coefficientssetintersectsetintersect(x,y)=intersection of the sets x and ysetminussetminus(x,y)=set of elements of x not belonging to ysetrandlLsetrand(n)=reset the seed of the random number generator to nsetsearchlGGD0,L,setsearch(x,y)=looks if y belongs to the set x. Returns 0 if it is not, otherwise returns the index j such that y==x[j]setunionsetunion(x,y)=union of the sets x and yshiftshift(x,n)=shift x left n bits if n>=0, right -n bits if n<0shiftmulshiftmul(x,n)=multiply x by 2^n (n>=0 or n<0)sigmasigma(x)=sum of the divisors of xsigmaksigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)iGsign(x)=sign of x, of type integer, real or fractionsignatsignat(x)=signature of the symmetric matrix xsignunitsignunit(bnf)=matrix of signs of the real embeddings of the system of fundamental units found by buchinitsimplefactmodsimplefactmod(x,p)=same as factmod except that only the degrees of the irreducible factors are givensimplifysimplify(x)=simplify the object x as much as possiblesinsin(x)=sine of xsinhsinh(x)=hyperbolic sine of xsizesize(x)=maximum number of decimal digits minus one of (the coefficients of) xsmallbasissmallbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallbuchinitsmallbuchinit(pol)=small buchinit, which can be converted to a big one using makebigbnfsmalldiscfsmalldiscf(x)=discriminant of the number field defined by the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallfactsmallfact(x)=partial factorization of the integer x (using only the stored primes)smallinitellsmallinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j]smallpolredsmallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)smallpolred2smallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)smithsmith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vectorsmith2smith2(x)=gives a three element vector [u,v,d] where u and v are square unimodular matrices such that d=u*x*v=diagonal(smith(x))smithclean(z)=if z=[u,v,d] as output by smith2, removes from u,v,d the rows and columns corresponding to entries equal to 1 in dsmithpolsmithpol(x)=Smith normal form (i.e. elementary divisors) of the matrix x with polynomial coefficients, expressed as a vectorsolvesolve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sortsort(x)=sort in ascending order of the vector xsqrsqr(x)=square of x. NOT identical to x*xsqredsqred(x)=square reduction of the (symmetric) matrix x ( returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)sqrtsqrt(x)=square root of xsrgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmsturm(x)=number of real roots of the polynomial xsturmpartsturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]LLDnsubcyclo(p,d)=finds an equation for the d-th degree subfield of Q(zeta_p), where p must be a prime powersubellsubell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve esubstGnGsubst(x,y,z)=in expression x, replace the variable y by the expression zsumsum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression exprsumaltsumalt(X=a,expr)=Villegas-Zagier's acceleration of alternating series expr, X starting at asumalt2sumalt2(X=a,expr)=Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at asuminfV=GIpsuminf(X=a,expr)=infinite sum (X goes from a to infinity) of real or complex expression exprsumpossumpos(X=a,expr)=sum of positive series expr, the formal variable X starting at asumpos2sumpos2(X=a,expr)=sum of positive series expr, the formal variable X starting at a, using Zagier's polynomialssupplementsupplement(x)=supplement the columns of the matrix x to an invertible matrixsylvestermatrix(x,y)=forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowstantan(x)=tangent of xtanhtanh(x)=hyperbolic tangent of xtaniyamaGPtaniyama(e)=modular parametrization of elliptic curve etaylorGnPtaylor(x,y)=taylor expansion of x with respect to the main variable of ytchebitchebi(n)=Tchebitcheff polynomial of degree n (n C-integer)teichteich(x)=teichmuller character of p-adic number xthetatheta(q,z)=Jacobi sine theta-functionthetanullkthetanullk(q,k)=k'th derivative at z=0 of theta(q,z)threetotwothreetotwo(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]threetotwo2threetotwo2(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]torselltorsell(e)=torsion subgroup of elliptic curve e: order, structure, generatorstracetrace(x)=trace of xtranstrans(x)=x~=transpose of xtrunctrunc(x)=truncation of x;when x is a power series,take away the O(X^)tschirnhaus(x)=random Tschirnhausen transformation of the polynomial xtwototwotwototwo(nf,a,b)=returns a 3-component vector [d,e,U] such that U is a unimodular 2x2 matrix with algebraic integer coefficients such that [a,b]*U=[d,e] and d,e are hopefully smallerunitunit(x)=fundamental unit of the quadratic field of discriminant x where x must be positiveuntil(a,seq)=evaluate the expression sequence seq until a is nonzerovaluationvaluation(x,p)=valuation of x with respect to pvecvec(x)=transforms the object x into a vector. Used mainly if x is a polynomial or a power seriesvecindexsortvecindexsort(x): indirect sorting of the vector xveclexsortveclexsort(x): sort the elements of the vector x in ascending lexicographic ordervecmaxvecmax(x)=maximum of the elements of the vector/matrix xvecminvecmin(x)=minimum of the elements of the vector/matrix xvecsort(x,k)=sorts the vector of vector (or matrix) x according to the value of its k-th componentvectorvector(n,X,expr)=row vector with n components of expression expr (X ranges from 1 to n)vvectorvvector(n,X,expr)=column vector with n components of expression expr (X ranges from 1 to n)weipellweipell(e)=formal expansion in x=z of Weierstrass P functionwfweberf(x)=Weber's f function of x (j=(f^24-16)^3/f^24)wf2weberf2(x)=Weber's f2 function of x (j=(f2^24+16)^3/f2^24)while(a,seq)=while a is nonzero evaluate the expression sequence seq. Otherwise 0zellzell(e,z)=In the complex case, lattice point corresponding to the point z on the elliptic curve ezetazeta(s)=Riemann zeta function at szetakzetak(nfz,s)=Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)zideallogzideallog(nf,x,bid)=if bid is a big ideal as given by zidealstarinit or zidealstarinitgen , gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)zidealstarzidealstar(nf,I)=3-component vector v, giving the structure of (Z_K/I)^*. v[1] is the order (i.e. phi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorszidealstarinitzidealstarinit(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*zidealstarinitgenzidealstarinitgen(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*znstarznstar(n)=3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. phi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorsthis function has been suppressedget_sep: argument too long (< %ld chars)integer too largearguments must be positive integerstex2mail -TeX -noindent -ragged -by_par realprecision = %ld significant digits (%ld digits displayed)realprecisiondefault: incorrect value for %s [%lu-%lu]significant termsseriesprecision %s = %lu %s %s = %lu format = %c%ld.%ld %c%ld.%lddefault: inexistent formatdarkbg1, 5, 3, 7, 6, 2, 3lightbg1, 6, 3, 4, 5, 2, 3boldfg[1,,1], [5,,1], [3,,1], [7,,1], [6,,1], , [2,,1][%ld,,%ld] colors = "%s" [%ld,%ld,%ld](use old functions, ignore case)(no backward compatibility)(warn when using obsolete functions)(use old functions, don't ignore case)compatibleuser functions re-initializedsecure[secure mode]: Do you want to modify the 'secure' flag? (^C if not) debug(bits 0x2/0x4 control matched-insert/arg-complete)readline01debugfilesdebugmemecholineshistsize(off)(on)(on with colors)(TeX output)PARIbreak\hskip 0pt plus \hsize\relax\discretionary{}{}{}}\ifx\%s\undefined \def\%s{%s}\fi PARIpromptSTART\vskip\medskipamount\bgroup\bfPARIpromptEND\egroup\bgroup\ttPARIinputEND\egroupPARIout\vskip\smallskipamount$\displaystyle{\tt\%#1} = #2$\ifx\%s\undefined \def\%s#1#2{%s}\fi [logfile was "%s"] alogfile(bits 0x2/0x4 control output of \left/\PARIbreak)TeXstyle(prettymatrix)(raw)(prettyprint)(external prettyprint)outputparisizeprimelimitstrictmatchtimer %s = "%s" factor_add_primesnew_galois_formatpsfilenone help = "%s" help[secure mode]: can't modify '%s' default (to %s) datadir = "%s" path = "%s" yes prettyprinter = "%s" broken prettyprinter: '%s'prettyprinter prompt%s = "%s" _contcomment> colorsdatadirformatpathpromptprompt_contunknown default: %sGPHELP/usr/local/bin/gphelpI was expecting an integer here %s = 1 (on) %s = 0 (off) default: incorrect value for %s [0:off / 1:on]expected character: ']'can't deflateunexpected characterunknown function or error in formal parametersvariable name expectedobsolete functionerror opening invalid flagWarning:Warning: increasing precWarning: failed toaccuracy problemsbug insorry,sorry, not yet available on this systemcollecting garbage inprecision too lowincorrect typeinconsistent dataimpossible assignment I-->Simpossible assignment I-->Ithe PARI stack overflows !length (lg) overflowexponent (expo) overflowvaluation (valp) overflowoverflow in R->dbl conversionnon invertible matrix in gaussnot a square matrixunknown identifier valence, please reportnot an integer argument in an arithmetic functionnot enough precomputed primesnot enough precomputed primes, need primelimit ~ impossible inverse modulo: constant polynomialnot a polynomialreducible polynomialzero polynomialtoo many iterations for desired precision in integration routinenot a definite matrix in lllgrambad argument for an elliptic curve related functionimpossibleforbiddendivision by zerotrying to overwrite a universal objectnot enough memoryinfinite precisionnegative exponentnon quadratic residue in gsqrtwhat's going on ?---- (type RETURN to continue) ----gp_readvec_stream: reaching %ld entries gp_readvec_stream: found %ld entries %c[%ld;%ld;%ldm%c[%ld;%ldm%c[0mCOLUMNSLINESout of range in integer -> character conversion (%ld)stndrdtht_INTt_REALt_INTMODt_FRACt_COMPLEXt_PADICt_QUADt_POLMODt_POLt_SERt_RFRACunknown type %ldt_QFRt_QFIt_VECt_COLt_MATt_LISTt_STRt_VECSMALL[&=%08lx] ,CLONE%s(lg=%ld%s):%08lx int = pol = mod = real = imag = p : p^l : I : coef of degree %ld = num = den = mat(%ld,%ld) = %ld%s column = %ld%s component = (lgeflist=%ld):(%c,varn=%ld,prec=%ld,valp=%ld):(%c,varn=%ld):(precp=%ld,valp=%ld):(%c,expo=%ld):(%c,lgefint=%ld):chars:gen_0 NULL %s : hash = %3ld, valence = %3ld, menu = %2ld, code = %s NULL next = %s %3ld:%3ld *** hashcode = %lu invalid range in print_functions_hashno such function Top : %lx Bottom : %lx Current stack : %lx Used : %ld long words (%ld K) Available : %ld long words (%ld K) Occupation of the PARI stack : %6.2f percent %ld objects on heap occupy %ld long words %ld variable names used out of %d ] ] [;] mod /+ O( 1)^%ldI + - #<%d> / } [] Vecsmall([])List() \frac{}{}^{%ld}\cdot () \*\right) \left( +\PARIbreak \pmatrix{ \cr} \cr \pmatrix{ \cr I/O: new pariFILE %s (code %d) I/O: closing file %s (code %d) deletecloseclose pipeI/O: opening file %s (mode %s) could not open requested file %stempfile %s already existsI/O: removed file %s I/O: can't remove file %srun-away comment. Closing itrun-away string. Closing itI/O: leaked file descriptor (%d): %sr[pipe:] '%s' failed.:~:~/gp%s/%sinput%s.gpYou never gave me anything to read!%s is a GP binary file. Please use writebinwrite failedunknown code in readobjsetting %s read failedmalformed binary file (no name) -%s not written for a %ld bit architectureunexpected endianness in %s%s written by an incompatible version of GPbinary output%s is not a GP binary file%ld unnamed objects read. Returning then in a vector[secure mode]: about to write to '%s'. OK ? (^C if not) I can't see into the future%s is set (%s), but is not a directory%s is set (%s), but is not writeableGPTMPDIR.%ld.%ld%.8s%scouldn't find a suitable name for a tempfile (%s)TMPDIR/tmp/var/tmpcouldn't find a suitable name for a tempdir (%s)unexpected closing braceembedded braces (in parser)[+++]0.E%ld0.TeX variable name too longthis object uses debugging variablesmod(Mod(qfr(qfi(Qfb(mat(Mat(matrix(0,%ld)matrix(0,%ld,j,k,0)[;]List([ I/O: checking output pipe... %s%sunknown user can't expand ~undefined environment variable: %s.Z.gz/usr/local/bin/gzip -dc%s %sskipping directory %sE%ldnew bloc, size %6lu (no %ld): %08lx mallocing NULL object in newblockilling bloc (no %ld): %08lx popping %s (bloc no %ld) mallocing NULL objectpari.pspari.logGP_DATA_DIR/usr/local/share/parixResetting all trapsCannot initialize kernelnot enough memory, new stack %lu *** Error in the PARI system. End of program. variable out of range in reorderduplicate indeterminates in reorderleaving recover() entering recover(), loc = %ld *** %s: no such error number: %ldcan't trap memory errors *** %s: %s. %s: %s in %s; new prec = %ld *** %suse pari_warn for warnings current stack size: %lu (%.3f Mbytes) [hint] you can increase GP stack with allocatemem() %s, please report %s is not yet implemented.%lu.additionmultiplication-->assignment %s %s %s %s.division in %s. ### user error: %s fileuncaught error: %ld For full compatibility with GP 1.39.15, type "default(compatible,3)", or set "compatible = 3" in your GPRC fileerrpiletypeergdiverinvmoderaccurerarchersigintertalkeruserthis trap keywordsignificant pointers lost in gerepile! (please report)bad component %ld in object %Zlbot>ltop in gerepiledoubling stack size; new stack = %lu (%.3f Mbytes)bot=0x%lx top=0x%lx 0x%p: 0x%lx %lu Time : %ld ColCol({x=[]}): transforms the object x into a column vector. Empty vector if x is omittedEulerEuler=Euler(): Euler's constant with current precisionI=I(): square root of -1ListList({x=[]}): transforms the vector or list x into a list. Empty list if x is omittedMatMat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is omittedGGD0,L,Mod(x,y): creates 'x modulo y'.O(a^b): p-adic or power series zero with precision given by bPiPi=Pi(): the constant pi, with current precisionPolGDnPol(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientPolrevPolrev(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termGGGDGpQfb(a,b,c,{D=0.}): binary quadratic form a*x^2+b*x*y+c*y^2. D is optional (0.0 by default) and initializes Shanks's distance if b^2-4*a*c>0SerSer(x,{v=x}): convert x (usually a vector) into a power series with variable v, starting with the constant coefficientSetSet({x=[]}): convert x into a set, i.e. a row vector with strictly increasing coefficients. Empty set if x is omittedStrStr({str}*): concatenates its (string) argument into a single stringStrchrStrchr(x): converts x to a string, translating each integer into a characterStrexpandStrexpand({str}*): concatenates its (string) argument into a single string, performing tilde expansionStrtexStrtex({str}*): translates its (string) arguments to TeX format and returns the resulting stringVecVec({x=[]}): transforms the object x into a vector. Empty vector if x is omittedVecsmallVecsmall({x=[]}): transforms the object x into a VECSMALL. Empty vector if x is omittedabs(x): absolute value (or modulus) of xacos(x): inverse cosine of xacosh(x): inverse hyperbolic cosine of xaddhelpvSsaddhelp(symbol,"message"): add/change help message for a symboladdprimes({x=[]}): add primes in the vector x to the prime table to be used in trial division. x may also be a single integer. Composite "primes" are allowed, and in that case you may later get a message "impossible inverse", which will give you some factors. List the current extra primes if x is omitted. If some primes are added which intersect non trivially the existing table entries, suitable updating is doneagm(x,y): arithmetic-geometric mean of x and yGLD0,L,palgdep(x,n,{flag=0}): algebraic relations up to degree n of x, using lindep([1,x,...,x^(n-1)], flag).aliasvrralias("new","old"): new is now an alias for oldallocatememvD0,L,allocatemem({s=0}): allocates a new stack of s bytes. doubles the stack if s is omittedarg(x): argument of x,such that -pi0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if S contains all primes less than 12.log(disc(Bnf))^2, where Bnf is the Galois closure of bnfbnfisprincipalGGD1,L,bnfisprincipal(bnf,x,{flag=1}): bnf being output by bnfinit (with flag<=2), gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. flag is optional, whose meaning is: 0: output only v; 1: default; 2: output only v, precision being doubled until the result is obtained; 3: as 2 but output generatorsbnfissunit(bnf,sfu,x): bnf being output by bnfinit (with flag<=2), sfu by bnfsunit, gives the column vector of exponents of x on the fundamental S-units and the roots of unity if x is a unit, the empty vector otherwisebnfisunitbnfisunit(bnf,x): bnf being output by bnfinit (with flag<=2), gives the column vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisebnfmake(sbnf): transforms small sbnf as output by bnfinit with flag=3 into a true big bnfbnfnarrowbnfnarrow(bnf): given a big number field as output by bnfinit, gives as a 3-component vector the structure of the narrow class groupbnfregbnfreg(P,{tech=[]}): compute the regulator of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfsignunitbnfsignunit(bnf): matrix of signs of the real embeddings of the system of fundamental units found by bnfinitbnfsunit(bnf,S): compute the fundamental S-units of the number field bnf output by bnfinit, S being a list of prime ideals. res[1] contains the S-units, res[5] the S-classgroup. See manual for detailsbnfunitbnfunit(bnf): compute the fundamental units of the number field bnf output by bnfinit when they have not yet been computed (i.e. with flag=2)bnrL1GDGD0,L,pbnrL1(bnr, {subgroup}, {flag=0}): bnr being output by bnrinit(,,1) and subgroup being a square matrix defining a congruence subgroup of bnr (the trivial subgroup if omitted), for each character of bnr trivial on this subgroup, compute L(1, chi) (or equivalently the first non-zero term c(chi) of the expansion at s = 0). The binary digits of flag mean 1: if 0 then compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is the order of L(s, chi) at s = 0, or if 1 then compute the value at s = 1 (and in this case, only for non-trivial characters), 2: if 0 then compute the value of the primitive L-function associated to chi, if 1 then compute the value of the L-function L_S(s, chi) where S is the set of places dividing the modulus of bnr (and the infinite places), 3: return also the charactersbnrclass(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this module. flag is optional, and can be 0: default, 1: compute data necessary for working in the ray class group, for example with functions such as bnrisprincipal or bnrdisc, without computing the generators of the ray class group, or 2: with the generators. When flag=1 or 2, the fifth component is the ray class group structure obtained when flag=0bnrclassnobnrclassno(bnf,x): ray class number of the module x for the big number field bnf. Faster than bnrclass if only the ray class number is wantedbnrclassnolist(bnf,list): if list is as output by ideallist or similar, gives list of corresponding ray class numbersbnrconductorGDGDGDGbnrconductor(a1,{a2},{a3},{flag=0}): conductor f of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc). flag is optional and can be 0: default, 1: returns [f, Cl_f, H], H subgroup of the ray class group modulo f defining the extension, 2: returns [f, bnr(f), H]bnrconductorofcharbnrconductorofchar(bnr,chi): conductor of the character chi on the ray class group bnrbnrdiscGDGDGD0,L,bnrdisc(a1,{a2},{a3},{flag=0}): absolute or relative [N,R1,discf] of the field defined by a1,a2,a3. [a1,{a2},{a3}] is of type [bnr], [bnr,subgroup], [bnf, module] or [bnf,module,subgroup], where bnf is as output by bnfclassunit (with flag<=2), bnr by bnrclass (with flag>0), and subgroup is the HNF matrix of a subgroup of the corresponding ray class group (if omitted, the trivial subgroup). flag is optional whose binary digits mean 1: give relative data; 2: return 0 if module is not the conductorbnrdisclistbnrdisclist(bnf,bound,{arch}): gives list of discriminants of ray class fields of all conductors up to norm bound, in a long vector The ramified Archimedean places are given by arch; all possible values are taken if arch is omitted. Supports the alternative syntax bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch (with units)bnrinit(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data linked to the ray class group structure corresponding to this module. flag is optional, and can be 0: default (same as bnrclass with flag = 1), 1: compute also the generators (same as bnrclass with flag = 2). The fifth component is the ray class group structurebnrisconductorlGDGDGbnrisconductor(a1,{a2},{a3}): returns 1 if the modulus is the conductor of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc), and 0 otherwise. Slightly faster than bnrconductor if this is the only desired resultbnrisprincipalbnrisprincipal(bnr,x,{flag=1}): bnr being output by bnrinit, gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. If (optional) flag is set to 0, output only vbnrrootnumberbnrrootnumber(bnr,chi,{flag=0}); returns the so-called Artin Root Number, i.e. the constant W appearing in the functional equation of the Hecke L-function associated to chi. Set flag = 1 if the character is known to be primitivebnrstarkbnrstark(bnr,{subgroup}): bnr being as output by bnrinit(,,1), finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the trivial subgroup if omitted) using Stark's units. The ground field and the class field must be totally real.breakD1,L,break({n=1}): interrupt execution of current instruction sequence, and exit from the n innermost enclosing loopsceil(x): ceiling of x=smallest integer>=xcenterlift(x,{v}): centered lift of x. Same as lift except for integermodschangevar(x,y): change variables of x according to the vector yGDnD0,L,charpoly(A,{v=x},{flag=0}): det(v*Id-A)=characteristic polynomial of the matrix or polmod A. flag is optional and may be set to 1 (use Lagrange interpolation) or 2 (use Hessenberg form), 0 being the defaultGDGchinese(x,{y}): x,y being both intmods (or polmods) computes z in the same residue classes as x and ycomponentcomponent(x,s): the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]. For list objects such as nf, bnf, bnr or ell, it is much easier to use member functions starting with "."concat(x,{y}): concatenation of x and y, which can be scalars, vectors or matrices, or lists (in this last case, both x and y have to be lists). If y is omitted, x has to be a list or row vector and its elements are concatenatedconj(x): the algebraic conjugate of xconjvec(x): conjugate vector of the algebraic number xcontent(x): gcd of all the components of x, when this makes sensecontfracGDGD0,L,contfrac(x,{b},{lmax}): continued fraction expansion of x (x rational,real or rational function). b and lmax are both optional, where b is the vector of numerators of the continued fraction, and lmax is a bound for the number of terms in the continued fraction expansioncontfracpnqncontfracpnqn(x): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction xcore(n,{flag=0}): unique (positive of negative) squarefree integer d dividing n such that n/d is a square. If (optional) flag is non-null, output the two-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisc(n,{flag=0}): discriminant of the quadratic field Q(sqrt(n)). If (optional) flag is non-null, output a two-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercos(x): cosine of xcosh(x): hyperbolic cosine of xcotan(x): cotangent of xdefaultD"",r,D"",s,D0,L,default({opt},{v}): returns the current value of the current default opt. If v is present, set opt to v first. If no argument is given, print a list of all defaults as well as their values.denominatordenominator(x): denominator of x (or lowest common denominator in case of an array)deriv(x,{y}): derivative of x with respect to the main variable of y, or to the main variable of x if y is omitteddilog(x): dilogarithm of xdirdiv(x,y): division of the Dirichlet series x by the Dirichlet series yV=GGEDGdireuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s). If c is present, output only the first c termsdirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet series ydirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1divisors(x): gives a vector formed by the divisors of x in increasing orderdivremGGDndivrem(x,y,{v}): euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainder, with respect to v (to main variable if v is omitted)eint1(x,{n}): exponential integral E1(x). If n is present, computes the vector of the first n values of the exponential integral E1(n.x) (x > 0)elladdelladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve eellakellak(e,n): computes the n-th Fourier coefficient of the L-function of the elliptic curve eellanellan(e,n): computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<2^24 on a 32-bit machine)ellapellap(e,p,{flag=0}): computes a_p for the elliptic curve e using Shanks-Mestre's method. flag is optional and can be set to 0 (default) or 1 (use Jacobi symbols)ellbilellbil(e,z1,z2): canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsellchangecurveellchangecurve(x,y): change data on elliptic curve according to y=[u,r,s,t]ellchangepointellchangepoint(x,y): change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]ellconvertnameellconvertname(name): convert an elliptic curve name (as found in the elldata database) from a string to a triplet [conductor, isogeny class, index]. It will also convert a triplet back to a curve name.elleisnumelleisnum(om,k,{flag=0}): om=[om1,om2] being a 2-component vector giving a basis of a lattice L and k an even positive integer, computes the numerical value of the Eisenstein series of weight k. When flag is non-zero and k=4 or 6, this gives g2 or g3 with the correct normalizationelletaelleta(om): om=[om1,om2], returns the two-component row vector [eta1,eta2] of quasi-periods associated to [om1,om2]ellgeneratorsellgenerators(E): if E is an elliptic curve as output by ellinit(), return the generators of the Mordell-Weil group associated to the curve. This function depends on the curve being referenced in the elldata database.ellglobalredellglobalred(e): e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pellheightGGD2,L,pellheight(e,x,{flag=2}): canonical height of point x on elliptic curve E defined by the vector e. flag is optional and selects the algorithm used to compute the archimedean local height. Its meaning is 0: use theta-functions, 1: use Tate's method, 2: use Mestre's AGMellheightmatrixellheightmatrix(e,x): gives the height matrix for vector of points x on elliptic curve e using theta functionsellidentifyellidentify(E): look up the elliptic curve E in the elldata database and return [[N, M, ...], C] where N is the name of the curve in J. E. Cremona database, M the minimal model and C the coordinates change (see ellchangecurve).ellinitGD0,L,pellinit(x,{flag=0}): x being the vector [a1,a2,a3,a4,a6] defining the curve Y^2 + a1.XY + a3.Y = X^3 + a2.X^2 + a4.X + a6, gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,disc,j,[e1,e2,e3],w1,w2,eta1,eta2,area]. If the curve is defined over a p-adic field, the last six components are replaced by root,u^2,u,q,w,0. If optional flag is 1, omit them altogether. x can also be a string, in this case the coefficients of the curve with matching name are looked in the elldata database if available.ellisoncurveellisoncurve(e,x): true(1) if x is on elliptic curve e, false(0) if notelljellj(x): elliptic j invariant of xelllocalredelllocalred(e,p): e being an elliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the Kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pelllseriesGGDGpelllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1ellminimalmodelGD&ellminimalmodel(e,{&v}): return the standard minimal integral model of the rational elliptic curve e. Sets v to the corresponding change of variablesellorderellorder(e,p): order of the point p on the elliptic curve e over Q, 0 if non-torsionellordinateellordinate(e,x): y-coordinates corresponding to x-ordinate x on elliptic curve eellpointtozellpointtoz(e,P): lattice point z corresponding to the point P on the elliptic curve eellpowellpow(e,x,n): n times the point x on elliptic curve e (n in Z)ellrootnolGDGellrootno(e,{p=1}): root number for the L-function of the elliptic curve e. p can be 1 (default), global root number, or a prime p (including 0) for the local root number at pellsearchellsearch(N): if N is an integer, it is taken as a conductor else if N is a string, it can be a curve name ("11a1"), a isogeny class ("11a") or a conductor ("11"). Return all curves in the elldata database that match the property.ellsigmaellsigma(om,z,{flag=0}): om=[om1,om2], value of the Weierstrass sigma function of the lattice generated by om at z if flag = 0 (default). If flag = 1, arbitrary determination of the logarithm of sigma. If flag = 2 or 3, same but using the product expansion instead of theta seriesellsubellsub(e,z1,z2): difference of the points z1 and z2 on elliptic curve eelltaniyamaelltaniyama(e): modular parametrization of elliptic curve eelltorselltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-NagellellwpGDGD0,L,pPellwp(e,{z=x},{flag=0}): Complex value of Weierstrass P function at z on the lattice generated over Z by e=[om1,om2] (e as given by ellinit is also accepted). Optional flag means 0 (default), compute only P(z), 1 compute [P(z),P'(z)], 2 consider om as an elliptic curve and compute P(z) for that curve (identical to ellztopoint in that case). If z is omitted or is a simple variable, return formal expansion in zellzetaellzeta(om,z): om=[om1,om2], value of the Weierstrass zeta function of the lattice generated by om at zellztopointellztopoint(e,z): coordinates of point P on the curve e corresponding to the complex number zerfc(x): complementary error functionerrorvs*error("msg"): abort script with error message msgeta(x,{flag=0}): if flag=0, eta function without the q^(1/24), otherwise eta of the complex number x in the upper half plane intelligently computed using SL(2,Z) transformationseulerphieulerphi(x): Euler's totient function of xeval(x): evaluation of x, replacing variables by their valueexp(x): exponential of xfactor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)factorbackGDGDGfactorback(f,{e},{nf}): given a factorisation f, gives the factored object back. If this is a prime ideal factorisation you must supply the corresponding number field as last argument. If e is present, f has to be a vector of the same length, and we return the product of the f[i]^e[i]factorcantorfactorcantor(x,p): factorization mod p of the polynomial x using Cantor-Zassenhausfactorff(x,p,a): factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factorialfactorial(x): factorial of x (x C-integer), the result being given as a real numberfactorintfactorint(x,{flag=0}): factor the integer x. flag is optional, whose binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM (huge composites will be declared prime)factormod(x,p,{flag=0}): factorization mod p of the polynomial x using Berlekamp. flag is optional, and can be 0: default or 1: simple factormod, same except that only the degrees of the irreducible factors are givenfactornf(x,t): factorization of the polynomial x over the number field defined by the polynomial tGGLD0,L,factorpadic(x,p,r,{flag=0}): p-adic factorization of the polynomial x to precision r. flag is optional and may be set to 0 (use round 4) or 1 (use Buchmann-Lenstra)ffinit(p,n,{v=x}): monic irreducible polynomial of degree n over F_p[v]fibonaccifibonacci(x): fibonacci number of index x (x C-integer)floor(x): floor of x = largest integer<=xfor(X=a,b,seq): the sequence is evaluated, X going from a up to bfordiv(n,X,seq): the sequence is evaluated, X running over the divisors of nforellvVLLIforell(E,a,b,seq): execute seq for each elliptic curves E of conductor between a and b in the elldata database.forprime(X=a,b,seq): the sequence is evaluated, X running over the primes between a and bforstep(X=a,b,s,seq): the sequence is evaluated, X going from a to b in steps of s (can be a vector of steps)vV=GDGIforsubgroup(H=G,{bound},seq): execute seq for each subgroup H of the abelian group G (in SNF form), whose index is bounded by bound. H is given as a left divisor of G in HNF formforvec(x=v,seq,{flag=0}): v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1 if flag is zero or omitted. If flag = 1 (resp. flag = 2), restrict to increasing (resp. strictly increasing) sequencesfrac(x): fractional part of x = x-floor(x)galoisexport(gal,{flag}): gal being a galois field as output by galoisinit, output a string representing the underlying permutation group in GAP notation (default) or Magma notation (flag = 1)GGD0,L,Dngaloisfixedfield(gal,perm,{flag},{v=y}): gal being a galois field as output by galoisinit and perm an element of gal.group or a vector of such elements, return [P,x] such that P is a polynomial defining the fixed field of gal[1] by the subgroup generated by perm, and x is a root of P in gal expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2 return [P,x,F] where F is the factorization of gal.pol over the field defined by P, where the variable v stands for a root of Pgaloisidentifygaloisidentify(gal): gal being a galois field as output by galoisinit, output the isomorphism class of the underlying abstract group as a two-components vector [o,i], where o is the group order, and i is the group index in the GAP4 small group librarygaloisinitgaloisinit(pol,{den}): pol being a polynomial or a number field as output by nfinit defining a Galois extension of Q, compute the Galois group and all neccessary informations for computing fixed fields. den is optional and has the same meaning as in nfgaloisconj(,4)(see manual)galoisisabelian(gal,{flag=0}): gal being as output by galoisinit, return 0 if gal is not abelian, the HNF matrix of gal over gal.gen if flag=0, 1 if flag is 1, and the SNF of gal is flag=2galoispermtopol(gal,perm): gal being a galois field as output by galoisinit and perm a element of gal.group, return the polynomial defining the corresponding Galois automorphismGDGD0,L,Dngaloissubcyclo(N,H,{fl=0},{v}):Compute a polynomial (in variable v) defining the subfield of Q(zeta_n) fixed by the subgroup H of (Z/nZ)*. N can be an integer n, znstar(n) or bnrinit(bnfinit(y),[n,[1]],1). H can be given by a generator, a set of generator given by a vector or a HNF matrix (see manual). If flag is 1, output only the conductor of the abelian extension. If flag is 2 output [pol,f] where pol is the polynomial and f the conductor.galoissubfieldsGD0,L,Dngaloissubfields(G,{flags=0},{v}):Output all the subfields of G. flags have the same meaning as for galoisfixedfieldgaloissubgroupsgaloissubgroups(G):Output all the subgroups of Ggamma(x): gamma function at xgammahgammah(x): gamma of x+1/2 (x integer)gcd(x,{y}): greatest common divisor of x and y.getheap(): 2-component vector giving the current number of objects in the heap and the space they occupygetrand(): current value of random number seedgetstack(): current value of stack pointer avmagettime(): time (in milliseconds) since last call to gettimeglobal(x): declare x to be a global variablelGGDGhilbert(x,y,{p}): Hilbert symbol at p of x,y. If x,y are integermods or p-adic, p can be omittedhyperu(a,b,x): U-confluent hypergeometric functionidealadd(nf,x,y): sum of two ideals x and y in the number field defined by nfidealaddtooneidealaddtoone(nf,x,{y}): if y is omitted, when the sum of the ideals in the number field K defined by nf and given in the vector x is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1. Otherwise, x and y are ideals, and if they sum up to 1, find one element in each of them such that the sum is 1idealappr(nf,x,{flag=0}): x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other p. If (optional) flag is non-null x must be a prime ideal factorization with possibly zero exponentsidealchinese(nf,x,y): x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprime(nf,x,y): gives an element b in nf such that b. x is an integral ideal coprime to the integral ideal yGGGD0,L,idealdiv(nf,x,y,{flag=0}): quotient x/y of two ideals x and y in HNF in the number field nf. If (optional) flag is non-null, the quotient is supposed to be an integral ideal (slightly faster)idealfactor(nf,x): factorization of the ideal x given in HNF into prime ideals in the number field nfidealhnfidealhnf(nf,a,{b}): hermite normal form of the ideal a in the number field nf, whatever form a may have. If called as idealhnf(nf,a,b), the ideal is given as aZ_K+bZ_K in the number field K defined by nfidealintersect(nf,x,y): intersection of two ideals x and y in the number field defined by nfidealinv(nf,x,{flag=0}): inverse of the ideal x in the number field nf. If flag is omitted or set to 0, use the different. If flag is 1 do not use itGLD4,L,ideallist(nf,bound,{flag=4}): vector of vectors L of all idealstar of all ideals of norm<=bound. If (optional) flag is present, its binary digits are toggles meaning 1: give generators; 2: add units; 4: give only the ideals and not the bid.ideallistarch(nf,list,arch): list is a vector of vectors of of bid's as output by ideallist. Return a vector of vectors with the same number of components as the original list. The leaves give information about moduli whose finite part is as in original list, in the same order, and archimedean part is now arch. The information contained is of the same kind as was present in the input.ideallogideallog(nf,x,bid): if bid is a big ideal, as given by idealstar(nf,I,1) or idealstar(nf,I,2), gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)idealminidealmin(nf,ix,{vdir}): minimum of the ideal ix in the direction vdir in the number field nfGGGD0,L,pidealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the number field nf. If (optional) flag is non-nul, reduce the resultidealnorm(nf,x): norm of the ideal x in the number field nfidealpow(nf,x,n,{flag=0}): n-th power of the ideal x in HNF in the number field nf If (optional) flag is non-null, reduce the resultidealprimedecidealprimedec(nf,p): prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a. Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantidealprincipalidealprincipal(nf,x): returns the principal ideal generated by the algebraic number x in the number field nfidealredidealred(nf,x,{vdir=0}): LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealstar(nf,I,{flag=1}): gives the structure of (Z_K/I)^*. flag is optional, and can be 0: simply gives the structure as a 3-component vector v such that v[1] is the order (i.e. eulerphi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generators. If flag=1 (default), gives idealstarinit, i.e. a 6-component vector [I,v,fa,f2,U,V] where v is as above without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*. Finally if flag=2, same as with flag=1 except that the generators are also givenidealtwoelt(nf,x,{a}): two-element representation of an ideal x in the number field nf. If (optional) a is non-zero, first element will be equal to aidealval(nf,x,p): valuation at p given in idealprimedec format of the ideal x in the number field nfideleprincipalideleprincipal(nf,x): returns the principal idele generated by the algebraic number x in the number field nfif(a,seq1,seq2): if a is nonzero, seq1 is evaluated, otherwise seq2. seq1 and seq2 are optional, and if seq2 is omitted, the preceding comma can be omitted alsoimag(x): imaginary part of xincgam(s,x,{y}): incomplete gamma function. y is optional and is the precomputed value of gamma(s)incgamc(s,x): complementary incomplete gamma functionintcircV=GGEDGpintcirc(X=a,R,s,{tab}): numerical integration of s on the circle |z-a|=R, divided by 2*I*Pi. tab is as in intnum.intformalintformal(x,{y}): formal integration of x with respect to the main variable of y, or to the main variable of x if y is omittedintfouriercosV=GGGEDGpintfouriercos(X=a,b,x,s,{tab}): numerical integration from a to b of cos(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the cosine-Fourier transform if a=-infty and b=+infty.intfourierexpintfourierexp(X=a,b,x,s,{tab}): numerical integration from a to b of exp(-2*I*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the ordinary Fourier transform if a=-infty and b=+infty. Note the minus sign.intfouriersinintfouriersin(X=a,b,x,s,{tab}): numerical integration from a to b of sin(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the sine-Fourier transform if a=-infty and b=+infty.intfuncinitV=GGED0,L,D0,L,pintfuncinit(X=a,b,s,{flag=0},{m=0}): initialize tables for integrations from a to b using a weight s(X). Essential for integral transforms such as intmellininv, intlaplaceinv and intfourier, since it avoids recomputing all the time the same quantities. Must then be used with intmellininvshort (for intmellininv) and directly with intnum and not with the corresponding integral transforms for the others. See help for intnum for coding of a and b, and m is as in intnuminit. If flag is nonzero, assumes that s(-X)=conj(s(X)), which is twice faster.intlaplaceinvintlaplaceinv(X=sig,x,s,{tab}): numerical integration on the line real(z) = sig of s(z)exp(xz)dz/(2*I*Pi), i.e. inverse Laplace transform of s at x. tab is as in intnum.intmellininvintmellininv(X=sig,x,s,{tab}): numerical integration on the line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). tab is as in intnum. Use intmellininvshort if several values must be computed.intmellininvshortintmellininvshort(sig,x,tab): numerical integration on the line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). Compulsory table tab has been precomputed using the command intfuncinit(t=[[-1],sig[2]],[[1],sig[2]],s) (with possibly its two optional additional parameters), where sig[2] = 1 if not given. Orders of magnitude faster than intmellininv.intnum(X=a,b,s,{tab}): numerical integration of s from a to b with respect to X. a (and similarly b) is coded as follows. It can be a scalar: f is assumed to be C^infty at a. It can be a two component vector [a[1],a[2]], where a[1] is the scalar, and a[2] is the singularity exponent (in ]-1,0]), logs being neglected. It can be a one component vector [1] or [-1] meaning +infty or -infty, slowly decreasing functions. It can be a two component vector [[1], z] or [[-1], z], where [1] or [-1] indicates +infty or -infty and z is coded as follows. If z is zero, slowly decreasing. If z is real positive, exponentially decreasing, of the type exp(-zX). If z<-1, very slowly decreasing like X^(-z). If z is complex nonreal, real part is ignored and if z = r+I*s then if s>0, cosine oscillation exactly cos(sX), while if s<0, sine oscillation exactly sin(sX). If f is exponentially decreasing times oscillating function, you have a choice, but it is in general better to choose the oscillating part. Finally tab is either 0 (let the program choose the integration step), a positive integer m (choose integration step 1/2^m), or a table tab precomputed with intnuminit (depending on the type of interval: compact, semi-compact or R, very slow, slow, exponential, or cosine or sine-oscillating decrease).intnuminitintnuminit(a,b,{m=0}): initialize tables for integrations from a to b. See help for intnum for coding of a and b. Possible types: compact interval, semi-compact (one extremity at + or - infinity) or R, and very slowly, slowly or exponentially decreasing, or sine or cosine oscillating at infinities,intnuminitgenVGGED0,L,D0,L,pintnuminitgen(t,a,b,ph,{m=0},{flag=0}): initialize tables for integrations from a to b using abcissas ph(t) and weights ph'(t). Note that there is no equal sign after the variable name t since t always goes from -infty to +infty, but it is ph(t) which goes from a to b, and this is not checked. If flag = 1 or 2, multiply the reserved table length by 4^flag, to avoid corresponding error.intnumrombV=GGED0,L,pintnumromb(X=a,b,s,{flag=0}): numerical integration of s (smooth in ]a,b[) from a to b with respect to X. flag is optional and mean 0: default. s can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be plus or minus infinity (chosen suitably), but of same sign; 3: s has only limits at a or bintnumsteplpintnumstep(): gives the default value of m used by all intnum and sumnum routines, such that the integration step is 1/2^m.isfundamentalisfundamental(x): true(1) if x is a fundamental discriminant (including 1), false(0) if notlGDGD&ispower(x,{k},{&n}): true (1) if x is a k-th power, false (0) if not. If n is given and a k-th root was computed in the process, put that in n. If k is omitted, return the maximal k >= 2 such that x = n^k is a perfect power, or 0 if no such k exist.isprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0) if not. If flag is 0 or omitted, use a combination of algorithms. If flag is 1, the primality is certified by the Pocklington-Lehmer Test. If flag is 2, the primality is certified using the APRCL test.ispseudoprimeispseudoprime(x,{n}): true(1) if x is a strong pseudoprime, false(0) if not. If n is 0 or omitted, use BPSW test, otherwise use strong Rabin-Miller test for n randomly chosen basesissquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is given puts the exact square root there if it was computedissquarefree(x): true(1) if x is squarefree, false(0) if notkillvSkill(x): kills the present value of the variable or function x. Returns new value or 0kroneckerkronecker(x,y): kronecker symbol (x/y)lcm(x,{y}): least common multiple of x and y, i.e. x*y / gcd(x,y)length(x): number of non code words in x, number of characters for a stringlex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x=0, transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwise. If p=-1, finds a basis of the intersection with Z^n of the lattice spanned by the columns of x. If p=-2, finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsize(x): number of rows and columns of the vector/matrix x as a 2-vectormatsnf(x,{flag=0}): Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector d. Binary digits of flag mean 1: returns [u,v,d] where d=u*x*v, otherwise only the diagonal d is returned, 2: allow polynomial entries, otherwise assume x is integral, 4: removes all information corresponding to entries equal to 1 in dmatsolvematsolve(M,B): gaussian solution of MX=B (M matrix, B column vector)matsolvemodmatsolvemod(M,D,B,{flag=0}): one solution of system of congruences MX=B mod D (M matrix, B and D column vectors). If (optional) flag is non-null return all solutionsmatsupplementmatsupplement(x): supplement the columns of the matrix x to an invertible matrixmattransposemattranspose(x): x~=transpose of xmax(x,y): maximum of x and ymin(x,y): minimum of x and yminpoly(A,{v=x}): minimal polynomial of the matrix or polmod A.modreverse(x): reverse polymod of the polymod x, if it existsmoebiusmoebius(x): Moebius function of xnewtonpoly(x,p): Newton polygon of polynomial x with respect to the prime pnextnext({n=1}): interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loopsnextprime(x): smallest pseudoprime >= xnfalgtobasisnfalgtobasis(nf,x): transforms the algebraic number x into a column vector on the integral basis nf.zkGD0,L,DGnfbasis(x,{flag=0},{p}): integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithm. Second and third args are optional. Binary digits of flag mean 1: assume that no square of a prime>primelimit divides the discriminant of x, 2: use round 2 algorithm instead. If present, p provides the matrix of a partial factorization of the discriminant of x, useful if one wants only an order maximal at certain primes onlynfbasistoalgnfbasistoalg(nf,x): transforms the column vector x on the integral basis into an algebraic numbernfdetint(nf,x): multiple of the ideal determinant of the pseudo generating set xnfdiscnfdisc(x,{flag=0},{p}): discriminant of the number field defined by the polynomial x using round 4. Optional args flag and p are as in nfbasisnfeltdivnfeltdiv(nf,a,b): element a/b in nfnfeltdiveucnfeltdiveuc(nf,a,b): gives algebraic integer q such that a-bq is smallnfeltdivmodprnfeltdivmodpr(nf,a,b,pr): element a/b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltdivremnfeltdivrem(nf,a,b): gives [q,r] such that r=a-bq is smallnfeltmodnfeltmod(nf,a,b): gives r such that r=a-bq is small with q algebraic integernfeltmulnfeltmul(nf,a,b): element a. b in nfnfeltmulmodprnfeltmulmodpr(nf,a,b,pr): element a. b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltpownfeltpow(nf,a,k): element a^k in nfnfeltpowmodprnfeltpowmodpr(nf,a,k,pr): element a^k modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltreducenfeltreduce(nf,a,id): gives r such that a-r is in the ideal id and r is smallnfeltreducemodprnfeltreducemodpr(nf,a,pr): element a modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltvalnfeltval(nf,a,pr): valuation of element a at the prime pr as output by idealprimedecnffactornffactor(nf,x): factor polynomial x in number field nfnffactormodnffactormod(nf,pol,pr): factorize polynomial pol modulo prime ideal pr in number field nfnfgaloisapplynfgaloisapply(nf,aut,x): Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfnfgaloisconj(nf,{flag=0},{den}): list of conjugates of a root of the polynomial x=nf.pol in the same number field. flag is optional (set to 0 by default), meaning 0: use combination of flag 4 and 1, always complete; 1: use nfroots; 2 : use complex numbers, LLL on integral basis (not always complete); 4: use Allombert's algorithm, complete if the field is Galois of degree <= 35 (see manual for detail). nf can be simply a polynomial with flag 0,2 and 4, meaning: 0: use combination of flag 4 and 2, not always complete (but a warning is issued when the list is not proven complete); 2 & 4: same meaning and restrictions. Note that only flag 4 can be applied to fields of large degrees (approx. >= 20)nfhilbertlGGGDGnfhilbert(nf,a,b,{p}): if p is omitted, global Hilbert symbol (a,b) in nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z) in nf, -1 otherwise. Otherwise compute the local symbol modulo the prime ideal pnfhnfnfhnf(nf,x): if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhnfmodnfhnfmod(nf,x,detx): if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfinit(pol,{flag=0}): pol being a nonconstant irreducible polynomial, gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]. flag is optional and can be set to 0: default; 1: do not compute different; 2: first use polred to find a simpler polynomial; 3: outputs a two-element vector [nf,Mod(a,P)], where nf is as in 2 and Mod(a,P) is a polymod equal to Mod(x,pol) and P=nf.pol; 4: as 2 but use a partial polred; 5: is to 3 what 4 is to 2nfisidealnfisideal(nf,x): true(1) if x is an ideal in the number field nf, false(0) if notnfisinclnfisincl(x,y): tests whether the number field x is isomorphic to a subfield of y (where x and y are either polynomials or number fields as output by nfinit). Return 0 if not, and otherwise all the isomorphisms. If y is a number field, a faster algorithm is usednfisisomnfisisom(x,y): as nfisincl but tests whether x is isomorphic to ynfkermodpr(nf,x,pr): kernel of the matrix x in Z_K/pr, where pr is in modpr format (see nfmodprinit)nfmodprinitnfmodprinit(nf,pr): transform the 5 element row vector pr representing a prime ideal into modpr format necessary for all operations mod pr in the number field nf (see manual for details about the format)nfnewprecnfnewprec(nf): transform the number field data nf into new data using the current (usually larger) precisionnfrootsDGGnfroots({nf},pol): roots of polynomial pol belonging to nf (Q if omitted) without multiplicitynfrootsof1nfrootsof1(nf): number of roots of unity and primitive root of unity in the number field nfnfsnfnfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfsolvemodpr(nf,a,b,pr): solution of a*x=b in Z_K/pr, where a is a matrix and b a column vector, and where pr is in modpr format (see nfmodprinit)nfsubfieldsnfsubfields(nf,{d=0}): find all subfields of degree d of number field nf (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nfnorm(x): norm of xnorml2(x): square of the L2-norm of the vector xnumbpartnumbpart(x): number of partitions of xnumdiv(x): number of divisors of xnumeratornumerator(x): numerator of xnumtopermnumtoperm(n,k): permutation number k (mod n!) of n letters (n C-integer)omega(x): number of distinct prime divisors of xpadicappr(x,a): p-adic roots of the polynomial x congruent to a mod ppadicprec(x,p): absolute p-adic precision of object xpermtonum(vect): ordinal (between 1 and n!) of permutation vectpolcoeff(x,s,{v}): coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[]). With respect to the main variable if v is omitted, with respect to the variable v otherwisepolcompositumpolcompositum(pol1,pol2,{flag=0}): vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2. If (optional) flag is set (i.e non-null), output for each compositum, not only the compositum polynomial pol, but a vector [pol,al1,al2,k] where al1 (resp. al2) is a root of pol1 (resp. pol2) expressed as a polynomial modulo pol, and a small integer k such that al2+k*al1 is the chosen root of polpolcyclopolcyclo(n,{v=x}): n-th cyclotomic polynomial (in variable v)poldegreelGDnpoldegree(x,{v}): degree of the polynomial or rational function x with respect to main variable if v is omitted, with respect to v otherwise. For scalar x, return 0 is x is non-zero and a negative number otherwisepoldiscpoldisc(x,{v}): discriminant of the polynomial x, with respect to main variable if v is omitted, with respect to v otherwisepoldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fpolgaloispolgalois(x): Galois group of the polynomial x (see manual for group coding). Return [n, s, k, name] where n is the order, s the signature, k the index and name is the GAP4 name of the transitive group.polhenselliftGGGLpolhensellift(x, y, p, e): lift the factorization y of x modulo p to a factorization modulo p^e using Hensel lift. The factors in y must be pairwise relatively prime modulo ppolinterpolateGDGDGD&polinterpolate(xa,{ya},{x},{&e}): polynomial interpolation at x according to data vectors xa, ya (ie return P such that P(xa[i]) = ya[i] for all i). If ya is omitter, return P such that P(i) = xa[i]. If present, e will contain an error estimate on the returned valuepolisirreduciblepolisirreducible(x): true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantpollead(x,{v}): leading coefficient of polynomial or series x, or x itself if x is a scalar. Error otherwise. With respect to the main variable of x if v is omitted, with respect to the variable v otherwisepollegendrepollegendre(n,{v=x}): legendre polynomial of degree n (n C-integer), in variable vpolrecip(x): reciprocal polynomial of xpolred(x,{flag=0},{p}): reduction of the polynomial x (gives minimal polynomials only). Second and third args are optional. The following binary digits of flag are significant 1: partial reduction, 2: gives also elements. p, if present, contains the complete factorization matrix of the discriminantpolredabs(x,{flag=0}): a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 norm. flag is optional, whose binary digit mean 1: give the element whose characteristic polynomial is the given polynomial. 4: give all polynomials of minimal T2 norm (give only one of P(x) and P(-x)). 16: partial reductionpolredordpolredord(x): reduction of the polynomial x, staying in the same orderGGDnD0,L,polresultant(x,y,{v},{flag=0}): resultant of the polynomials x and y, with respect to the main variables of x and y if v is omitted, with respect to the variable v otherwise. flag is optional, and can be 0: default, assumes that the polynomials have exact entries (uses the subresultant algorithm), 1 for arbitrary polynomials, using Sylvester's matrix, or 2: using a Ducos's modified subresultant algorithmpolroots(x,{flag=0}): complex roots of the polynomial x. flag is optional, and can be 0: default, uses Schonhage's method modified by Gourdon, or 1: uses a modified Newton methodpolrootsmod(x,p,{flag=0}): roots mod p of the polynomial x. flag is optional, and can be 0: default, or 1: use a naive search, useful for small ppolrootspadicpolrootspadic(x,p,r): p-adic roots of the polynomial x to precision rpolsturmpolsturm(x,{a},{b}): number of real roots of the polynomial x in the interval]a,b] (which are respectively taken to be -oo or +oo when omitted)polsubcyclopolsubcyclo(n,d,{v=x}): finds an equation (in variable v) for the d-th degree subfields of Q(zeta_n). Output is a polynomial or a vector of polynomials is there are several such fields, or none.polsylvestermatrixpolsylvestermatrix(x,y): forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowspolsym(x,n): vector of symmetric powers of the roots of x up to npoltchebipoltchebi(n,{v=x}): Tchebitcheff polynomial of degree n (n C-integer), in variable vpoltschirnhauspoltschirnhaus(x): random Tschirnhausen transformation of the polynomial xLGD0,L,ppolylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of xpolzagierpolzagier(n,m): Zagier's polynomials of index n,mprecision(x,{n}): change the precision of x to be n (n C-integer). If n is omitted, output real precision of object xprecprimeprecprime(x): largest pseudoprime <= x, 0 if x<=1prime(n): returns the n-th prime (n C-integer)primepi(x): the prime counting function pi(x) = #{p <= x, p prime}.primes(n): returns the vector of the first n primes (n C-integer)printprint(a): outputs a (in raw format) ending with newlineprint1print1(a): outputs a (in raw format) without ending with newlineprintpprintp(a): outputs a (in beautified format) ending with newlineprintp1printp1(a): outputs a (in beautified format) without ending with newlineprinttexprinttex(a): outputs a in TeX formatprod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of expressionV=GGEpprodeuler(X=a,b,expr): Euler product (X runs over the primes between a and b) of real or complex expressionV=GED0,L,pprodinf(X=a,expr,{flag=0}): infinite product (X goes from a to infinity) of real or complex expression. flag can be 0 (default) or 1, in which case compute the product of the 1+expr insteadpsi(x): psi-function at xqfbclassno(x,{flag=0}): class number of discriminant x using Shanks's method by default. If (optional) flag is set to 1, use Euler productsqfbcomprawqfbcompraw(x,y): Gaussian composition without reduction of the binary quadratic forms x and yqfbhclassnoqfbhclassno(x): Hurwitz-Kronecker class number of x>0qfbnucompqfbnucomp(x,y,l): composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedqfbnupowqfbnupow(x,n): n-th power of primitive positive definite quadratic form x using nucomp and nuduplqfbpowrawqfbpowraw(x,n): n-th power without reduction of the binary quadratic form xqfbprimeformqfbprimeform(x,p): returns the prime form of discriminant x, whose first coefficient is pGD0,L,DGDGDGqfbred(x,{flag=0},{D},{isqrtD},{sqrtD}): reduction of the binary quadratic form x. All other args. are optional. D, isqrtD and sqrtD, if present, supply the values of the discriminant, floor(sqrt(D)) and sqrt(D) respectively. If D<0, its value is not used and all references to Shanks's distance hereafter are meaningless. flag can be any of 0: default, uses Shanks's distance function d; 1: use d, do a single reduction step; 2: do not use d; 3: do not use d, single reduction step.qfbsolve(Q,p): Return [x,y] so that Q(x,y)=p where Q is a binary quadratic form and p a prime number, or 0 if there is no solution.qfgaussredqfgaussred(x): square reduction of the (symmetric) matrix x (returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)qfjacobiqfjacobi(x): eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xqflll(x,{flag=0}): LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix). The columns of x must be linearly independent, unless specified otherwise below. flag is optional, and can be 0: default, 1: assumes x is integral, columns may be dependent, 2: assumes x is integral, returns a partially reduced basis, 4: assumes x is integral, returns [K,I] where K is the integer kernel of x and I the LLL reduced image, 5: same as 4 but x may have polynomial coefficients, 8: same as 0 but x may have polynomial coefficientsqflllgram(x,{flag=0}): LLL reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix). flag is optional and can be 0: default,1: lllgramint algorithm for integer matrices, 4: lllgramkerim giving the kernel and the LLL reduced image, 5: lllgramkerimgen same when the matrix has polynomial coefficients, 8: lllgramgen, same as qflllgram when the coefficients are polynomialsGDGDGD0,L,pqfminim(x,{bound},{maxnum},{flag=0}): number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0. flag is optional, and can be 0: default; 1: returns the first minimal vector found (ignore maxnum); 2: as 0 but uses a more robust, slower implementation, valid for non integral quadratic formsqfperfectionqfperfection(a): rank of matrix of xx~ for x minimal vectors of a gram matrix aqfrepqfrep(x,B,{flag=0}): vector of (half) the number of vectors of norms from 1 to B for the integral and definite quadratic form x. Binary digits of flag mean 1: count vectors of even norm from 1 to 2B, 2: return a t_VECSMALL instead of a t_VECqfsignqfsign(x): signature of the symmetric matrix xquadclassunitquadclassunit(D,{flag=0},{tech=[]}): compute the structure of the class group and the regulator of the quadratic field of discriminant D. If flag is non-null (and D>0), compute the narrow class group. See manual for the optional technical parametersquaddisc(x): discriminant of the quadratic field Q(sqrt(x))quadgen(x): standard generator of quadratic order of discriminant xquadhilbertquadhilbert(D,{pq}): relative equation for the Hilbert class field of the quadratic field of discriminant D (which can also be a bnf). If D<0, pq (if supplied) is a 2-component vector [p,q], where p,q are the prime numbers needed for Schertz's method. In that case, return 0 if [p,q] not suitable.quadpoly(D,{v=x}): quadratic polynomial corresponding to the discriminant D, in variable vquadray(D,f,{lambda}): relative equation for the ray class field of conductor f for the quadratic field of discriminant D (which can also be a bnf). For D < 0, lambda (if supplied) is the technical element of bnf necessary for Schertz's method. In that case, return 0 if lambda is not suitable.quadregulatorquadregulator(x): regulator of the real quadratic field of discriminant xquadunitquadunit(x): fundamental unit of the quadratic field of discriminant x where x must be positiverandom({N=2^31}): random integer between 0 and N-1readvecD"",s,readvec({filename}): create a vector whose components are the evaluation of all the expressions found in the input file filenamereal(x): real part of xremoveprimesremoveprimes({x=[]}): remove primes in the vector x (with at most 100 components) from the prime table. x can also be a single integer. List the current extra primes if x is omittedreorder({x=[]}): reorder the variables for output according to the vector x. If x is void or omitted, print the current list of variablesreturnreturn({x=0}): return from current subroutine with result xrnfalgtobasisrnfalgtobasis(rnf,x): relative version of nfalgtobasis, where rnf is a relative numberfieldrnfbasis(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfbasistoalgrnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a relative numberfieldrnfcharpolyGGGDnrnfcharpoly(nf,T,alpha,{var=x}): characteristic polynomial of alpha over nf, where alpha belongs to the algebra defined by T over nf. Returns a polynomial in variable var (x by default)rnfconductor(bnf,polrel,{flag=0}): conductor of the Abelian extension of bnf defined by polrel. The result is [conductor,rayclassgroup,subgroup], where conductor is the conductor itself, rayclassgroup the structure of the corresponding full ray class group, and subgroup the HNF defining the norm group (Artin or Takagi group) on the given generators rayclassgroup[3]. If flag is non-zero, check that polrel indeed defines an Abelian extensionrnfdedekind(nf,T,pr): relative Dedekind criterion over nf, applied to the order defined by a root of irreducible polynomial T, modulo the prime ideal pr. Returns [flag,basis,val], where basis is a pseudo-basis of the enlarged order, flag is 1 iff this order is pr-maximal, and val is the valuation in pr of the order discriminantrnfdetrnfdet(nf,order): given a pseudomatrix, compute its pseudodeterminantrnfdiscrnfdisc(nf,pol): given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfeltabstorelrnfeltabstorel(rnf,x): transforms the element x from absolute to relative representationrnfeltdownrnfeltdown(rnf,x): expresses x on the base field if possible; returns an error otherwisernfeltreltoabsrnfeltreltoabs(rnf,x): transforms the element x from relative to absolute representationrnfeltuprnfeltup(rnf,x): expresses x (belonging to the base field) on the relative fieldrnfequation(nf,pol,{flag=0}): given a pol with coefficients in nf, gives the absolute equation apol of the number field defined by pol. flag is optional, and can be 0: default, or non-zero, gives [apol,th], where th expresses the root of nf.pol in terms of the root of apolrnfhnfbasisrnfhnfbasis(bnf,order): given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfidealabstorel(rnf,x): transforms the ideal x from absolute to relative representationrnfidealdownrnfidealdown(rnf,x): finds the intersection of the ideal x with the base fieldrnfidealhnfrnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a relative numberfieldrnfidealmulrnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a relative numberfieldrnfidealnormabsrnfidealnormabs(rnf,x): absolute norm of the ideal xrnfidealnormrelrnfidealnormrel(rnf,x): relative norm of the ideal xrnfidealreltoabsrnfidealreltoabs(rnf,x): transforms the ideal x from relative to absolute representationrnfidealtwoeltrnfidealtwoelt(rnf,x): relative version of idealtwoelt, where rnf is a relative numberfieldrnfidealuprnfidealup(rnf,x): lifts the ideal x (of the base field) to the relative fieldrnfinitrnfinit(nf,pol): pol being a non constant irreducible polynomial defined over the number field nf, initializes a vector of data necessary for working in relative number fields (rnf functions). See manual for technical detailsrnfisfree(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnfisnorm(T,x,{flag=0}): T is as output by rnfisnorminit applied to L/K. Tries to tell whether x is a norm from L/K. Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-integer, with S a list of places in K containing the ramified primes, generators of the class group of ext, as well as those primes dividing x. If L/K is Galois, omit flag, otherwise it is used to add more places to S: all the places above the primes p <= flag (resp. p | flag) if flag > 0 (resp. flag < 0). The answer is guaranteed (i.e x norm iff b=1) if L/K is Galois or, under GRH, if S contains all primes less than 12.log(disc(M))^2, where M is the normal closure of L/KGGD2,L,rnfisnorminit(pol,polrel,{flag=2}): let K be defined by a root of pol, L/K the extension defined by polrel. Compute technical data needed by rnfisnorm to solve norm equations Nx = a, for x in L, and a in K. If flag=0, do not care whether L/K is Galois or not; if flag = 1, assume L/K is Galois; if flag = 2, determine whether L/K is Galoisrnfkummerrnfkummer(bnr,{subgroup},{deg=0}): bnr being as output by bnrinit, finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the ray class field if subgroup is omitted). deg can be zero (default), or positive, and in this case the output is the list of all relative equations of degree deg for the given bnrrnflllgram(nf,pol,order): given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfnormgroup(bnr,polrel): norm group (or Artin or Takagi group) corresponding to the Abelian extension of bnr.bnf defined by polrel, where the module corresponding to bnr is assumed to be a multiple of the conductor. The result is the HNF defining the norm group on the given generators in bnr[5][3]rnfpolred(nf,pol): given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simplerrnfpolredabs(nf,pol,{flag=0}): given a pol with coefficients in nf, finds a relative simpler polynomial defining the same field. Binary digits of flag mean: 1: return also the element whose characteristic polynomial is the given polynomial, 2: return an absolute polynomial, 16: partial reductionrnfpseudobasis(nf,pol): given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitz(nf,order): given an order as output by rnfpseudobasis, gives [A,I,D,d] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialround(x,{&e}): take the nearest integer to all the coefficients of x. If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitsserconvolserconvol(x,y): convolution (or Hadamard product) of two power seriesserlaplaceserlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of a_n*x^n. For the reverse operation, use serconvol(x,exp(X))serreverseserreverse(x): reversion of the power series xsetintersect(x,y): intersection of the sets x and ysetissetsetisset(x): true(1) if x is a set (row vector with strictly increasing entries), false(0) if notsetminus(x,y): set of elements of x not belonging to ysetrand(n): reset the seed of the random number generator to nsetsearch(x,y,{flag=0}): looks if y belongs to the set x. If flag is 0 or omitted, returns 0 if it is not, otherwise returns the index j such that y==x[j]. If flag is non-zero, return 0 if y belongs to x, otherwise the index j where it should be insertedsetunion(x,y): union of the sets x and yshift(x,n): shift x left n bits if n>=0, right -n bits if n<0.shiftmul(x,n): multiply x by 2^n (n>=0 or n<0)GD1,L,sigma(x,{k=1}): sum of the k-th powers of the divisors of x. k is optional and if omitted is assumed to be equal to 1sign(x): sign of x, of type integer, real or fractionsimplify(x): simplify the object x as much as possiblesin(x): sine of xsinh(x): hyperbolic sine of xsizebytesizebyte(x): number of bytes occupied by the complete tree of the object xsizedigitsizedigit(x): maximum number of decimal digits minus one of (the coefficients of) xsolve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sqr(x): square of x. NOT identical to x*xsqrt(x): square root of xsqrtintsqrtint(x): integer square root of x (x integer)sqrtnGGD&psqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is set to a suitable root of unity to recover all solutions. If it was not possible, z is set to zerosubgrouplist(bnr,{bound},{flag=0}): bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G, outputs the list of subgroups of G (of index bounded by bound, if not omitted), given as HNF left divisors of the SNF matrix corresponding to G. If flag=0 (default) and bnr is as output by bnrinit, gives only the subgroups for which the modulus is the conductorsubst(x,y,z): in expression x, replace the variable y by the expression zsubstpolsubstpol(x,y,z): in expression x, replace the polynomial y by the expression z, using remainder decomposition of x.substvec(x,v,w): in expression x, make a best effort to replace the variables v1,...,vn by the expression w1,...,wnsum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of expression exprsumalt(X=a,expr,{flag=0}): Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialssumdivGVEsumdiv(n,X,expr): sum of expression expr, X running over the divisors of nV=GEpsuminf(X=a,expr): infinite sum (X goes from a to infinity) of real or complex expression exprsumnumV=GGEDGD0,L,psumnum(X=a,sig,expr,{tab},{flag=0}): numerical summation of expr from X = ceiling(a) to +infinity. sig is either a scalar or a two-component vector coding the function's decrease rate at infinity. It is assumed that the scalar part of sig is to the right of all poles of expr. If present, tab must be initialized by sumnuminit. If flag is nonzero, assumes that conj(expr(z)) = expr(conj(z)).sumnumaltsumnumalt(X=a,sig,s,{tab},{flag=0}): numerical summation of (-1)^X s from X = ceiling(a) to +infinity. Note that the (-1)^X must not be included. sig is either a scalar or a two-component vector coded as in intnum, and the scalar part is larger than all the real parts of the poles of s. Uses intnum, hence tab is as in intnum. If flag is nonzero, assumes that the function to be summed satisfies conj(f(z))=f(conj(z)), and then up to twice faster.sumnuminitGD0,L,D1,L,psumnuminit(sig, {m=0}, {sgn=1}): initialize tables for numerical summation. sgn is 1 (in fact >= 0), the default, for sumnum (ordinary sums) or -1 (in fact < 0) for sumnumalt (alternating sums). sig is as in sumnum and m is as in intnuminit.sumpos(X=a,expr,{flag=0}): sum of positive series expr, the formal variable X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialstan(x): tangent of xtanh(x): hyperbolic tangent of xtaylor(x,y): taylor expansion of x with respect to the main variable of yteichmullerteichmuller(x): teichmuller character of p-adic number xtheta(q,z): Jacobi sine theta-functionthetanullk(q,k): k'th derivative at z=0 of theta(q,z)thuethue(tnf,a,{sol}): solve the equation P(x,y)=a, where tnf was created with thueinit(P), and sol, if present, contains the solutions of Norm(x)=a modulo units in the number field defined by P. If tnf was computed without assuming GRH (flag 1 in thueinit), the result is unconditionalthueinitthueinit(P,{flag=0}): initialize the tnf corresponding to P, that will be used to solve Thue equations P(x,y) = some-integer. If flag is non-zero, certify the result unconditionnaly. Otherwise, assume GRH (much faster of course)trace(x): trace of xtrapD"",r,DIDItrap({err}, {rec}, {seq}): try to execute seq, trapping error err (all of them if err ommitted); sequence rec is executed if the error occurs and is the result of the command. When seq is omitted, define rec as a default handler for error err (a break loop will be started if rec omitted). If rec is the empty string "" pop out the last default handlertruncatetruncate(x,{&e}): truncation of x; when x is a power series,take away the O(X^). If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitstypetype(x): return the type of the GEN x.until(a,seq): evaluate the expression sequence seq until a is nonzerovaluation(x,p): valuation of x with respect to pvariablevariable(x): main variable of object x. Gives p for p-adic x, error for scalarsvecextractvecextract(x,y,{z}): extraction of the components of the matrix or vector x according to y and z. If z is omitted, y designs columns, otherwise y corresponds to rows and z to columns. y and z can be vectors (of indices), strings (indicating ranges as in "1..10") or masks (integers whose binary representation indicates the indices to extract, from left to right 1, 2, 4, 8, etc.)vecmax(x): maximum of the elements of the vector/matrix xvecmin(x): minimum of the elements of the vector/matrix xvecsort(x,{k},{flag=0}): sorts the vector of vectors (or matrix) x in ascending order, according to the value of its k-th component if k is not omitted. Binary digits of flag (if present) mean: 1: indirect sorting, return the permutation instead of the permuted vector, 2: sort using lexicographic order, 4: use descending instead of ascending orderGDVDIvector(n,{X},{expr=0}): row vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0svectorsmallvectorsmall(n,{X},{expr=0}): VECSMALL with n components of expression expr (X ranges from 1 to n) which must be small integers. By default, fill with 0svectorvvectorv(n,{X},{expr=0}): column vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0sweber(x,{flag=0}): One of Weber's f function of x. flag is optional, and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x) such that (j=(f^24-16)^3/f^24), 1: function f1(x)=eta(x/2)/eta(x) such that (j=(f1^24+16)^3/f2^24), 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x) such that (j=(f2^24+16)^3/f2^24)while(a,seq): while a is nonzero evaluate the expression sequence seq. Otherwise 0writevss*write(filename,a): write the string expression a (same output as print) to filenamewrite1write1(filename,a): write the string expression a (same output as print1) to filenamewritebinvsDGwritebin(filename,{x}): write x as a binary object to file filename. If x is omitted, write all session variableswritetexwritetex(filename,a): write the string expression a (same format as print) to filename, in TeX formatzeta(s): Riemann zeta function at s with s a complex or a p-adic numberzetak(nfz,s,{flag=0}): Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by zetakinit (NOT by nfinit) flag is optional, and can be 0: default, compute zetak, or non-zero: compute the lambdak function, i.e. with the gamma factorszetakinitzetakinit(x): compute number field information necessary to use zetak, where x is an irreducible polynomialGGGDGzncoppersmith(P, N, X, {B=N}): finds all integers x0 with |x0| <= X such that gcd(N, P(x0)) > B. X should be smaller than exp((log B)^2 / (deg(P) log N)).znlogznlog(x,g): g as output by znprimroot (modulo a prime). Return smallest non-negative n such that g^n = xznorderznorder(x,{o}): order of the integermod x in (Z/nZ)*. Optional o is assumed to be a multiple of the order.znprimrootznprimroot(n): returns a primitive root of n when it existsznstar(n): 3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. eulerphi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorsuser interruptfloating point exception: bug in PARI or calling programbus error: bug in PARI or calling programsegmentation fault: bug in PARI or calling programunknown signalbroken pipebad object %Zincorrect abscissa in sumnumincorrect a or b in intnumboth nonzero real and imag. part in coding, real ignoredm too large in intnuminitintnuminit0sumnuminit0incorrect table length in intnum initializationinfinities of the same sign in intnuminitgeninfinities of different type in intnuminitgenneed exponential decrease in intinvmellinshortqrom3: iteration %ld: %Z qrom2: iteration %ld: %Z integral from infty to infty or from -infty to -inftycode error in intnumintegral transformexponential increase in integral transformx = 0 in FourierFourier transform of oscillating functionsincorrect beginning value in sumnumray regulator.furay unitsray torsion unitscurve not defined over Rcurve not defined over a p-adic fieldforparistep equal to zero in forstepprime_loop_initnot a vector of two-component vectors in forvecnot a vector in forvecnon integral index in sumnon integral index in suminfnon integral index in prodinfnon integral index in prodinf1constant term != 1 in direulernegative number of components in vectornegative number of rows in matrixnegative number of columns in matrixidentical index variables in matrixnon integral index in sumaltnon integral index in sumposnon integral index in sumpos2roots must be bracketed in solvetoo many iterations in solveIndividual Fermat powerings: %-3ld: %3ld Number of Fermat powerings = %lu Maximal number of nondeterministic steps = %lu Step6: testing potential divisors Step5: testing conditions lp Jacobi sums and tables computed Step4: q-values (# = %ld, largest = %ld): aprcl: e(t) too smallChoosing t = %ld aprcl test fails! this is highly improbableIncorrect curve name in ellconvertnameIncorrect vector in ellconvertname%s/elldata/ell%ldElliptic files %s not compatible Elliptic curves files not available for conductor %ld [missing %s]No such elliptic curveIncorrect curve name in ellsearchIncomplete curve name in ellsearchNo such elliptic curve in databaseXY%Z - (%Z) powell for non integral, non CM, exponentsbadgoodellpointtoz: %s square root z = %Z z1 = %Z z2 = %Z w1 and w2 R-linearly dependent in elliptic functionk not a positive even integer in elleisnumreduction mod SL2 (reduce_z)can't evaluate ellzeta at a polecan't evaluate log(ellsigma) at lattice pointexpecting a simple variable in ellwpnot an integral curve in elllocalrednot a prime in localredprime too large in apell2, use apell[apell1] giant steps, i = %ld[apell1] sorting[apell1] baby steps, s = %ldnot a prime in apellnot an integral modelanell for n >= %lunot an integral model in akellcut-off point must be positive in lseriesellpoint not on elliptic curvetwo vector/matrix types in bilhellorderell for nonrational elliptic curvesellinit data not accurate enough. Increase precisionsingular curve in ellinitincompatible p-adic numbers in initellpowell for nonintegral CM exponentnot a complex multiplication in powellnorm too large in CMCM_ellpowweipellnumlocalred (nu_D - nu_j != 0,6)localred (p | c6)not a rational curve in ellintegralmodelapell (f^(i*s) = 1)%lu is not prime, use ellaktorsell (bug1)torsell (bug3)torsell (bug2)valuation of j must be negative in p-adic ellinitinitell for 2-adic numbersPartitions of %ld (%ld) i = %ld: %Z partitions( %ld ) is meaningless Output of isin_%ld_G_H(%ld,%ld): %ld Reordering of the roots: ( %d ) Output of isin_%ld_G_H(%ld,%ld): not included. %s/galdata/NAM%ldgalois files %s not compatible Galois names files not available, please upgrade galdata [missing %s]galois in degree > 11too large precision in preci()Galoisbig: reduced polynomial #1 = %Z discriminant = %Z ODDEVEN%s group %ld^%ld %2ld: %Z $$$$$ New prec = %ld $$$$$ Tschirnhaus transformation of degree %ld: $$$$$ tschirn all integer roots are double roots Working with polynomial #%ld: # rational integer roots = %ld:more than %ld rational integer roots ----> Group # %ld/%ld: RES%s/galdata/%s%ld_%ld_%ldopening %sgalois files not available [missing %s] *** Entering isin_%ld_G_H_(%ld,%ld) read_objectincorrect value in bin()COS_%ldindefinite invariant polynomial in gpoly()Not a group in group_identGaloisIndex: Using hash value w=%ld GaloisIndex: Using hash value u=%ld GaloisIndex: Using hash value s=%ld galoisindex for groups of order >127Classification of transitive groups of order > 30 is not knownnot an n-th power in idealsqrtnbug%d in kummerpolrel(beta) = %Z beta reduced = %Z [rnfkummer] candidate listStep 18 Step 16 Step 14, 15 and 17 Step 13 Step 12 Step 9, 10 and 11 Step 8 Step 6 Step 5 Step 4 [rnfkummer] Selmer group[rnfkummer] bnfinit(Kz)Step 3 [rnfkummer] compositumStep 2 polred(compositum) = %Z Step 1 kummer for composite relative degree[rnfkummer] conductormain variable in kummer must not be xreducebetabeta LLL-reduced mod U^l = %Z beta reduced via ell-th root = %Z reducing beta = %Z isvirtualunit%ld(%ld) Computing Newton sums: naive reduction mod U^l: unit exp. = %Z MPQS: found factor = %ld whilst creating factor base MPQS: sizing out of tune, FB size or tolerance too largeMPQSLPTMPFRELFNEWLPRELLPNEWCOMBMPQS: found %lu candidate%s MPQS: passing the %3.1f%% sort point, time = %ld ms MPQS: no factors found. MPQS: time in Gauss and gcds = %ld ms MPQS: restarting sieving ... MPQS: giving up. MPQS: found %ld factors = %Z%s MPQS: found factors = %Z and %Z MPQS: found factor = %Z MPQS: starting Gauss over F_2 on %ld relations and combiningMPQS: done sorting%s, time = %ld ms MPQS: found %3.1f%% of the required relations MPQS: found %ld full relations MPQS: Net yield: %4.3g full relations per 100 candidates MPQS: %4.3g full relations per 100 polynomials MPQS: %4.1f%% of the polynomials yielded no candidates MPQS: next sort point at %3.1f%% MPQS: (%ld of these from partial relations) MPQS: renamed file %s to %s cannot rename file %s to %s MPQS: split N whilst combining, time = %ld ms MPQS: passing the %3.1f%% sort point MPQS: Ran out of primes for A, giving up. MPQS: starting main loop MPQS: first sorting at %ld%%, then every %3.1f%% / %3.1f%% MPQS: sieve threshold = %u MPQS: sieving interval = [%ld, %ld] MPQS: size of factor base = %ld MPQS: striving for %ld relations MPQS: coefficients A will be built from %ld primes each MPQS: primes for A to be chosen near FB[%ld] = %ld MPQS: smallest prime used for sieving FB[%ld] = %ld MPQS: largest prime in FB = %ld MPQS: bound for `large primes' = %ld MPQS: sizing out of tune, FB too small or way too few primes in AMPQS: computing logarithm approximations for p_i in FB MPQS: creating factor base and allocating arrays... MPQS: number too big to be factored with MPQS, giving upMPQS: found multiplier %ld for N MPQS: factoring number of %ld decimal digits MPQS: number to factor N = %Z manyseveralMPQS: factoring this number will take %s hours: N = %ZMPQS: Gauss elimination will require more than 128MBy of memory (estimated memory needed: %4.1fMBy) MPQS: kN = %Z MPQS: kN has %ld decimal digits ,<%lu>,%lu,%lu...] Wait a second -- MPQS: FB [-1,2MPQS: precomputing auxiliary primes up to %ld error whilst writing to file %sMPQS: done sorting one file. MQPS: short of space -- another buffer for sorting MPQS: relations file truncated?! MQPS: line wrap -- another buffer for sorting error whilst flushing file %sMPQS: chose Q_%ld(x) = %Z x^2 %c %Z x + C %lu %lu%s @ %s :%s %s :%s %ld %ld : 0MPQS: combining {%ld @ %s : %s} * {%ld @ %s : %s} == {%s} 1 1MPQS: combined %ld full relation%s MPQS: Gauss done: kernel has rank %ld, taking gcds... , looking for more...MPQS: got %ld factors%s comp.unknown packaging %ld: %Z ^%ld (%s) MPQS: wrapping up vector of %ld factors [3]: mpqs_solve_linear_systemMPQS: resplitting a factor after %ld kernel vectors MPQS: splitting N after %ld kernel vector%s MPQS: got two factors, looking for more... MPQS: X^2 - Y^2 != 0 mod N index i = %ld MPQS: wrong relation found after Gauss[2]: mpqs_solve_linear_systemMPQS (relation is a nonsquare)[1]: mpqs_solve_linear_systemFREL file truncated?!cannot seek FREL fileMPQS: no solutions found from linear system solver\\ KERNEL COMPUTED BY MPQS KERNEL= [1, 0, \\ MATRIX READ BY MPQS FREL=MPQS: chose primes for A FB[%ld]=%ld%sMPQS: new bit pattern for primes for A: 0x%lX MPQS: wrapping, more primes for A now chosen near FB[%ld] = %ld ftell error on full relations fileshorterlongerMPQS: full relations file %s than expectedMPQS panickingMPQS: decomposed a square cube5th power7th powerMPQS: decomposed a %s polynomial variable must have highest priority in nffactormodpolynomial variable must have highest priority in nfrootstest if polynomial is square-free nfissplitpolynomial variable must have highest priority in nffactornumber of factor(s) found: %ld squarefree test Entering nffactor: nfsqffUsing Trager's method splitting mod %Zbound computationroot 1) T_2 bound for %s: %Z 2) Conversion from T_2 --> | |^2 bound : %Z 3) Final bound: %Z choice of a prime idealPrime ideal chosen: %Z incorrect variables in rnfcharpolynf_factor_boundfor this exponent, GSmin = %Z Time reduction: %ld exponent: %ld Hensel lift%3ld %s at prime %Z Time: %ld to find factor %Zremaining modular factor(s): %ld @nf_LLL_cmbffor this trace... lifted (avma - bot = %lu) ... mod p^k (avma - bot = %lu) nf_LLL_cmbf: checking factor %ld (avma - bot = %lu) partition functionarg to partition function must be < 10^15incorrect character in bnrrootnumberquickpolpolrelnumCompute %sCompute polrelnumAllStarkstark (computation impossible)Recpolnumpolrel = %Z polrelnum = %Z It's not a square... Checking the square-root of the Stark unit... zetavalues = %Z N0 in QuickPol: %ld Compute Wquadhilbertrealnew precision: %ld FindModulusCompute Cl(k)class field not totally real in bnrstarkincorrect subgrp in bnrstarkbase field not totally real in bnrstarkmain variable in bnrstark must not be xno non-trivial character in bnrL1incorrect subgroup in bnrL1the ground field must be distinct from QNo, we're done! Modulus = %Z and subgroup = %Z Trying to find another modulus... Trying modulus = %Z and subgroup = %Z Looking for a modulus of norm: ArtinNumberconductor too large in ArtinNumber* Root Number: cond. no %ld/%ld (%ld chars) RecCoeff (cf = %ld, B = %Z) S & T character no: %ld (%ld/%ld) * conductor no %ld/%ld (N = %ld) Init: N0 = %ld Not enough precomputed primes (need all p <= %ld)S&TCplxModuluscpl = 2^%ld diff(CHI) = %ZRecCoeffRecCoeff3: no solution found! TR_POL(1), i = %ld/%ldTR_POL(-1), i = %ld/%ldTR_POL, i = %ld/%ldoverflow in calc_blocklg(Z) = %ld, lg(Y) = %ld Z = %Z Y = %Z ns = %ld changing f(x): p divides disc(g) embedding = %Z candidate = %Z coeff too big for pol g(x) pol. found = %Z d-1 test failed delta[%ld] = %Z Subfields of degree %ld: %Z * Look for subfields of degree %ld coeff too big for embedding lifting embedding mod p^k = %Z^%ld Chosen prime: p = %ld p = %ld, lcm = %ld, orbits: %Z sorry, too many block systems in nfsubfieldsf = %Z p = %Z, lift to p^%ld 2 * Hadamard bound * ind = %Z 2 * M = %Z Entering compute_data() ***** Leaving subfields ***** Entering subfields pol = %Z prec = %d invalid polynomial in thue (need deg>2) c6 = %Z c8 = %Z c11 = %Z c15 = %Z - norm sol. no %ld/%ld get_emb* real root no %ld/%ld c10 = %Z c13 = %Z x1 -> %Z x2 -> %Z c14 = %Z Non trivial conditional class group. *** May miss solutions of the norm equationAll solutions are <= %Z not a tnf in thueexpected an integer in bnfisintnorm%Z eliminated because of sign looking for a fundamental unit of norm -1 invalid polynomial in thue (need n>2)inithueepsilon_3 -> %Z c1 = %Z c2 = %Z Indice <= %Z non-monic polynomial in thueNot enough precision in thueChecking (\pm %Z, \pm %Z) SmallSols* Checking for small solutions Semirat. reduction: B0 -> %Z thue (totally rational case)CF failed. Increasing kappa B0 -> %Z Entering CF... errdelta = %Z LLL failed. Increasing kappa Semirat. reduction: B0 -> %Z x <= %Z Entering LLL... sol = %Z Partial = %Z gcd f_P does not divide n_p B0 = %Z Baker = %Z C (bitsize) : %d LLL_First_Pass successful !! B0 -> %Z x <= %Z operatorlibparimemberintegerrefcardtutorialbnrellrnfmodulusGoodbye! %s%s \PARIpromptSTART|%s\PARIpromptEND|%s\PARIinputEND|%% %s %cverbatim:readline not availableBreak loop (type 'break' or Control-d to go back to GP)break> ? [type in empty line to continue] [secure mode]: system commands not allowed Tried to run '%s'no input ???... skipping file '%s' GP/PARI CALCULATOR Version 2.3.2 (released)i386 running darwin (ix86 kernel) 32-bit version3.3 20030304 (Apple Computer, Inc. build 1809)gcc-%sJan 19 2008compiled: %s, %snot compiled in not(readline %s, extended help%s available)Copyright (C) 2000-2006 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = %lu, primelimit = %lu compiled: %s### Errors on startup, exiting... %%%ld = %clatex:$\blue %s$%c\magenta\%%%ld = t_%s is aliased to: aide (no help found)new identifier (no valence assigned)%s: %s user defined variableinstalled functionunknown identifierHelp topics: for a list of relevant subtopics, type ?n for n in 0: user-defined identifiers (variable, alias, function) 1: Standard monadic or dyadic OPERATORS 2: CONVERSIONS and similar elementary functions 3: TRANSCENDENTAL functions 4: NUMBER THEORETICAL functions 5: Functions related to ELLIPTIC CURVES 6: Functions related to general NUMBER FIELDS 7: POLYNOMIALS and power series 8: Vectors, matrices, LINEAR ALGEBRA and sets 9: SUMS, products, integrals and similar functions 10: GRAPHIC functions 11: PROGRAMMING under GP 12: The PARI community Also: ? functionname (short on-line help) ?\ (keyboard shortcuts) ?. (member functions) Extended help looks available: ?? (opens the full user's manual in a dvi previewer) ?? tutorial / refcard / libpari (tutorial/reference card/libpari manual) ?? keyword (long help text about "keyword" from the user's manual) ??? keyword (a propos: list of related functions).# : enable/disable timer ## : print time for last result \\ : comment up to end of line \a {n} : print result in raw format (readable by PARI) \b {n} : print result in beautified format \c : list all commands (same effect as ?*) \d : print all defaults \e {n} : enable/disable echo (set echo=n) \g {n} : set debugging level \gf{n} : set file debugging level \gm{n} : set memory debugging level \h {m-n}: hashtable information \l {f} : enable/disable logfile (set logfile=f) \m {n} : print result in prettymatrix format \o {n} : change output method (0=raw, 1=prettymatrix, 2=prettyprint, 3=2-dim) \p {n} : change real precision \ps{n} : change series precision \q : quit completely this GP session \r {f} : read in a file \s {n} : print stack information \t : print the list of PARI types \u : print the list of user-defined functions \um : print the list of user-defined member functions \v : print current version of GP \w {nf} : write to a file \x {n} : print complete inner structure of result \y {n} : disable/enable automatic simplification (set simplify=n) {f}=optional filename. {n}=optional integer Member functions, followed by relevant objects a1-a6, b2-b8, c4-c6 : coeff. of the curve. ell area : area ell bid : big ideal bnr bnf : big number field bnf, bnr clgp : class group bid, bnf, bnr cyc : cyclic decomposition (SNF) bid, clgp, bnf, bnr diff, codiff: different and codifferent nf, bnf, bnr disc : discriminant ell, nf, bnf, bnr e, f : inertia/residue degree prid fu : fundamental units bnf, bnr gen : generators bid, prid, clgp, bnf, bnr index: index nf, bnf, bnr j : j-invariant ell mod : modulus bid, bnr nf : number field bnf, bnr no : number of elements bid, clgp, bnf, bnr omega, eta: [omega1,omega2] and [eta1, eta2] ell p : rational prime below prid prid pol : defining polynomial nf, bnf, bnr reg : regulator bnf, bnr roots: roots ell nf, bnf, bnr sign,r1,r2 : signature nf, bnf, bnr t2 : t2 matrix nf, bnf, bnr tate : Tate's [u^2, u, q] ell tu : torsion unit and its order bnf, bnr w : Mestre's w ell zk : integral basis nf, bnf, bnr no such section in help: ?The standard distribution of GP/PARI includes a reference manual, a tutorial, a reference card and quite a few examples. They should have been installed in the directory '%s'. If not, ask the person who installed PARI on your system where they can be found. You can also download them from the PARI WWW site 'http://pari.math.u-bordeaux.fr/' Three mailing lists are devoted to PARI: - pari-announce (moderated) to announce major version changes. - pari-dev for everything related to the development of PARI, including suggestions, technical questions, bug reports and patch submissions. - pari-users for everything else! To subscribe, send an empty message to -subscribe@list.cr.yp.to. An archive is kept at the WWW site mentioned above. You can also reach the authors directly by email: pari@math.u-bordeaux.fr (answer not guaranteed).setting %ld history entriesList of the PARI types: t_INT : long integers [ cod1 ] [ cod2 ] [ man_1 ] ... [ man_k ] t_REAL : long real numbers [ cod1 ] [ cod2 ] [ man_1 ] ... [ man_k ] t_INTMOD : integermods [ code ] [ mod ] [ integer ] t_FRAC : irred. rationals [ code ] [ num. ] [ den. ] t_COMPLEX: complex numbers [ code ] [ real ] [ imag ] t_PADIC : p-adic numbers [ cod1 ] [ cod2 ] [ p ] [ p^r ] [ int ] t_QUAD : quadratic numbers [ cod1 ] [ mod ] [ real ] [ imag ] t_POLMOD : poly mod [ code ] [ mod ] [ polynomial ] ------------------------------------------------------------- t_POL : polynomials [ cod1 ] [ cod2 ] [ man_1 ] ... [ man_k ] t_SER : power series [ cod1 ] [ cod2 ] [ man_1 ] ... [ man_k ] t_RFRAC : irred. rat. func. [ code ] [ num. ] [ den. ] t_QFR : real qfb [ code ] [ a ] [ b ] [ c ] [ del ] t_QFI : imaginary qfb [ code ] [ a ] [ b ] [ c ] t_VEC : row vector [ code ] [ x_1 ] ... [ x_k ] t_COL : column vector [ code ] [ x_1 ] ... [ x_k ] t_MAT : matrix [ code ] [ col_1 ] ... [ col_k ] t_LIST : list [ code ] [ cod2 ] [ x_1 ] ... [ x_k ] t_STR : string [ code ] [ man_1 ] ... [ man_k ] t_VECSMALL: vec. small ints [ code ] [ x_1 ] ... [ x_k ] serieslengthtime = %ld ms%ld,00%s%ldmn, %ldh, user interrupt after *** last result computed in GP (Floating Point Exception)GP (Bus Error)GP (Segmentation Fault)signal handlingBroken Pipe, resetting file stack...buffersize is no longer used. -b ignored### Usage: %s [options] [GP files] [-q,--quiet] Quiet mode: do not print banner and history numbersOptions are: [-p,--primelimit primelimit] Precalculate primes up to the limit [-s,--stacksize stacksize] Start with the PARI stack of given size (in bytes) [-f,--fast] Faststart: do not read .gprc [--emacs] Run as if in Emacs shell [--test] Test mode. No history, wrap long lines (bench only) [--texmacs] Run as if using TeXmacs frontend [--help] Print this message [--version] Output version info and exit [--version-short] Output version number and exit version-shortversiontexmacsemacstestquietfaststacksize%lu.%lu.%lu macsest \LITERALnoLENGTH{%s}\%%%ld =\LITERALnoLENGTH{%s} \PARIout{%ld}%s%%%ld = %s\%%%ld = @%d%s -fromgp %s %c%s%s%cugly_kludge_done-kno external help programreadmissing '='notDone. unknown preprocessor variableunknown directive...skipping line %ld. _QUOTE_BACKQUOTE_DOUBQUOTEGPRCHOME/etc/gprcC:/_gprcReading GPRC: %s ...HOMEDRIVEHOMEPATHEMACSREADLVERSIONexternextern(cmd): execute shell command cmd, and feeds the result to GP (as if loading from file)input(): read an expression from the input file or standard inputquitvquit(): quits GP and return to the systemread({filename}): read from the input file filename. If filename is omitted, reread last input file, be it from read() or \rsystemvssystem(a): a being a string, execute the system command a (not valid on every machine)whatnowvrwhatnow(fun): if fun was present in GP version 1.39.15 or lower, gives the new function namevLpallocatemem(s)=allocates a new stack of s bytes, or doubles the stack if size is 0boxvLGGbox(w,x2,y2)=if the cursor is at position (x1,y1), draw a box with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move)colorvLLcolor(w,c)=set default color to c in rectwindow. Possible values for c are 1=sienna, 2=cornsilk, 3=red, 4=black, 5=grey, 6=blue, 7=gainsboroughcursorcursor(w)=current position of cursor in rectwindow wdefault({opt},{v},{flag}): set the default opt to v. If v is omitted, print the current default for opt. If no argument is given, print a list of all defaults as well as their values. If flag is non-zero, return the result instead of printing it on screen. See manual for detailsdrawvGpdraw(list)=draw vector of rectwindows list at indicated x,y positions; list is a vector w1,x1,y1,w2,x2,y2,etc...initrectvLLLinitrect(w,x,y)=initialize rectwindow w to size x,ykill(x)=kills the present value of the variable or function x. Returns new value or 0killrectvLkillrect(w)=erase the rectwindow wlineline(w,x2,y2)=if cursor is at position (x1,y1), draw a line from (x1,y1) to (x2,y2) (and move the cursor) in the rectwindow wlines(w,listx,listy)=draws an open polygon in rectwindow w where listx and listy contain the x (resp. y) coordinates of the verticesmovemove(w,x,y)=move cursor to position x,y in rectwindow wplotvV=GGIDGDGpplot(X=a,b,expr)=crude plot of expression expr, X goes from a to bplothploth(X=a,b,expr)=plot of expression expr, X goes from a to b in high resolutionploth2ploth2(X=a,b,[expr1,expr2])=plot of points [expr1,expr2], X goes from a to b in high resolutionplothmultplothmult(X=a,b,[expr1,...])=plot of expressions expr1,..., X goes from a to b in high resolutionplothrawplothraw(listx,listy)=plot in high resolution points whose x (resp. y) coordinates are in listx (resp. listy)pointpoint(w,x,y)=draw a point (and move cursor) at position x,y in rectwindow wpointspoints(w,listx,listy)=draws in rectwindow w the points whose x (resp y) coordinates are in listx (resp listy)postdrawpostdraw(list)=same as plotdraw, except that the output is a PostScript program in file "pari.ps"postplothV=GGIpD0,L,D0,L,postploth(X=a,b,expr)=same as ploth, except that the output is a PostScript program in the file "pari.ps"postploth2V=GGIpD0,L,postploth2(X=a,b,[expr1,expr2])=same as ploth2, except that the output is a PostScript program in the file "pari.ps"postplothrawpostplothraw(listx,listy)=same as plothraw, except that the output is a PostScript program in the file "pari.ps"pprintpprint(a)=outputs a in beautified format ending with newlinepprint1pprint1(a)=outputs a in beautified format without ending with newlineprint(a)=outputs a in raw format ending with newlineprint1(a)=outputs a in raw format without ending with newlinerboxrbox(w,dx,dy)=if the cursor is at (x1,y1), draw a box with diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move)read()=read an expression from the input file or standard inputrlinerline(w,dx,dy)=if the cursor is at (x1,y1), draw a line from (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow wrlinesrlines(w,dx,dy)=draw in rectwindow w the points given by vector of first coordinates xsand vector of second coordinates, connect them by linesrmovermove(w,dx,dy)=move cursor to position (dx,dy) relative to the present position in the rectwindow wrpointrpoint(w,dx,dy)=draw a point (and move cursor) at position dx,dy relative to present position of the cursor in rectwindow wrpointsrpoints(w,xs,ys)=draw in rectwindow w the points given by vector of first coordinates xs and vector of second coordinates ysscalevLGGGGscale(w,x1,x2,y1,y2)=scale the coordinates in rectwindow w so that x goes from x1 to x2 and y from y1 to y2 (y20, or return the current precision if n<=0setserieslengthsetserieslength(n)=set the default length of power series to n if n>0, or return the current default length if n<=0settypesettype(x,t)=make a copy of x with type t (to use with extreme care)stringvLsstring(w,x)=draw in rectwindow w the string corresponding to x, where x is either a string, or a number in R, written in format 9.3texprinttexprint(a)=outputs a in TeX formattype(x)=internal type number of the GEN xcan't find symbol '%s' in library '%s'can't find symbol '%s' in dynamic symbol table of process%s couldn't open dynamic library '%s'couldn't open dynamic symbol table of process[secure mode]: about to install '%s'. OK ? (^C if not) installvrrD"",r,D"",s,install(name,code,{gpname},{lib}): load from dynamic library 'lib' the function 'name'. Assign to it the name 'gpname' in this GP session, with argument code 'code'. If 'lib' is omitted use 'libpari.so'. If 'gpname' is omitted, use 'name'plot(X=a,b,expr,{ymin},{ymax}): crude plot of expression expr, X goes from a to b, with Y ranging from ymin to ymax. If ymin (resp. ymax) is not given, the minima (resp. the maxima) of the expression is used insteadplotboxplotbox(w,x2,y2): if the cursor is at position (x1,y1), draw a box with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move)plotclipplotclip(w): clip the contents of the rectwindow to the bounding box (except strings)plotcolorplotcolor(w,c): in rectwindow w, set default color to c. Possible values for c are 1=black, 2=blue, 3=sienna, 4=red, 5=cornsilk, 6=grey, 7=gainsboroughplotcopyvLLGGD0,L,plotcopy(sourcew,destw,dx,dy,{flag=0}): copy the contents of rectwindow sourcew to rectwindow destw with offset (dx,dy). If flag's bit 1 is set, dx and dy express fractions of the size of the current output device, otherwise dx and dy are in pixels. dx and dy are relative positions of northwest corners if other bits of flag vanish, otherwise of: 2: southwest, 4: southeast, 6: northeast cornersplotcursorplotcursor(w): current position of cursor in rectwindow wplotdrawvGD0,L,plotdraw(list, {flag=0}): draw vector of rectwindows list at indicated x,y positions; list is a vector w1,x1,y1,w2,x2,y2,etc. . If flag!=0, x1, y1 etc. express fractions of the size of the current output deviceV=GGIpD0,M,D0,L, Parametric|1; Recursive|2; no_Rescale|4; no_X_axis|8; no_Y_axis|16; no_Frame|32; no_Lines|64; Points_too|128; Splines|256; no_X_ticks|512; no_Y_ticks|1024; Same_ticks|2048ploth(X=a,b,expr,{flags=0},{n=0}): plot of expression expr, X goes from a to b in high resolution. Both flags and n are optional. Binary digits of flags mean: 1=Parametric, 2=Recursive, 4=no_Rescale, 8=no_X_axis, 16=no_Y_axis, 32=no_Frame, 64=no_Lines (do not join points), 128=Points_too (plot both lines and points), 256=Splines (use cubic splines), 512=no_X_ticks, 1024= no_Y_ticks, 2048=Same_ticks (plot all ticks with the same length). n specifies number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplothraw(listx,listy,{flag=0}): plot in high resolution points whose x (resp. y) coordinates are in listx (resp. listy). If flag is 1, join points, other non-0 flags should be combinations of bits 8,16,32,64,128,256 meaning the same as for ploth()plothsizesD0,L,plothsizes({flag=0}): returns array of 6 elements: terminal width and height, sizes for ticks in horizontal and vertical directions, width and height of characters. If flag=0, sizes of ticks and characters are in pixels, otherwise are fractions of the screen sizeplotinitvLD0,G,D0,G,D0,L,plotinit(w,{x=0},{y=0},{flag=0}): initialize rectwindow w to size x,y. If flag!=0, x and y express fractions of the size of the current output device. x=0 or y=0 means use the full size of the deviceplotkillplotkill(w): erase the rectwindow wplotlinesvLGGD0,L,plotlines(w,listx,listy,{flag=0}): draws an open polygon in rectwindow w where listx and listy contain the x (resp. y) coordinates of the vertices. If listx and listy are both single values (i.e not vectors), draw the corresponding line (and move cursor). If (optional) flag is non-zero, close the polygonplotlinetypeplotlinetype(w,type): change the type of following lines in rectwindow w. type -2 corresponds to frames, -1 to axes, larger values may correspond to something else. w=-1 changes highlevel plottingplotmoveplotmove(w,x,y): move cursor to position x,y in rectwindow wplotpointsplotpoints(w,listx,listy): draws in rectwindow w the points whose x (resp y) coordinates are in listx (resp listy). If listx and listy are both single values (i.e not vectors), draw the corresponding point (and move cursor)plotpointsizevLGplotpointsize(w,size): change the "size" of following points in rectwindow w. w=-1 changes global valueplotpointtypeplotpointtype(w,type): change the type of following points in rectwindow w. type -1 corresponds to a dot, larger values may correspond to something else. w=-1 changes highlevel plottingplotrboxplotrbox(w,dx,dy): if the cursor is at (x1,y1), draw a box with diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move)plotrecthLV=GGIpD0,L,D0,L,plotrecth(w,X=xmin,xmax,expr,{flags=0},{n=0}): plot graph(s) for expr in rectwindow w, where expr is scalar for a single non-parametric plot, and a vector otherwise. If plotting is parametric, its length should be even and pairs of entries give points coordinates. If not, all entries but the first are y-coordinates. Both flags and n are optional. Binary digits of flags mean: 1 parametric plot, 2 recursive plot, 4 do not rescale w, 8 omit x-axis, 16 omit y-axis, 32 omit frame, 64 do not join points, 128 plot both lines and points. n specifies the number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplotrecthrawLGD0,L,plotrecthraw(w,data,{flags=0}): plot graph(s) for data in rectwindow w, where data is a vector of vectors. If plot is parametric, length of data should be even, and pairs of entries give curves to plot. If not, first entry gives x-coordinate, and the other ones y-coordinates. Admits the same optional flags as plotrecth, save that recursive plot is meaninglessplotrlineplotrline(w,dx,dy): if the cursor is at (x1,y1), draw a line from (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow wplotrmoveplotrmove(w,dx,dy): move cursor to position (dx,dy) relative to the present position in the rectwindow wplotrpointplotrpoint(w,dx,dy): draw a point (and move cursor) at position dx,dy relative to present position of the cursor in rectwindow wplotscaleplotscale(w,x1,x2,y1,y2): scale the coordinates in rectwindow w so that x goes from x1 to x2 and y from y1 to y2 (y2 %s%s New syntax: %s%s ===> %s%s this function was suppressed blackbluevioletredredgreengreygainsboro9x15cannot open 9x15 font *** X fatal error: %s rectplotWM_DELETE_WINDOWWM_PROTOCOLSno X serverlost display on %s%9.*g%s %10s%-9.7g%*.7g incorrect rectwindow number in graphic function (%ld not in [0, %ld])incorrect dimensions in initrectyou must initialize the rectwindow firstThis is not a valid colorrectpointsrectlines%%! 50 50 translate /p {moveto 0 2 rlineto 2 0 rlineto 0 -2 rlineto closepath fill} def /l {lineto} def /m {moveto} def /Times-Roman findfont %ld scalefont setfont %g %g scale stroke showpage postscriptgtodblListsingle vector in gtodblListinconsistent data in rectplothinflag PLOT_PARAMETRIC ignorednot a row vector in plothmulti-curves cannot be plot recursively%.5gnot a vector in rectdrawrectdrawincorrect number of components in rectdraw%ld %ld p %ld %ld m %ld %ld l stroke %ld %ld m %ld %ld l %ld %ld l %ld %ld l closepath %ld %ld m %ld %ld l (\)) %ld %ld m 90 rotate show -90 rotate (%sToo few points (%ld) for spline plot 008@@@@@@``` P@@@N $$6666HHHQlDD (2<Pddxxx@@@h  @8@8p_7-c}qeaUCWIi$ syo(7#7;89I-1_e<1QQuEF5 K?OT/ S5K;X%uI = aZ{c[g'h__[' 3 inq Q~E  A Y9 dF uwuous, ## 7h3KSKv+vuKKV4+uuus#LsK CÐ vsv#MsL#xw 0LL'vvvܮvvv|v\<:{<{\ 5l5L5[vvvۮL,xwwwL52||{ w)P\!!6(fO (4(x)uU545P-+$H=`5;w'xw* z |)57Ǡvxyxby*x6j6J *7|"7*yzZzBWZt s j(J*(t+)Bx|6~<9P*#ʮ‘¯w xjwRwzwwJw>po@#ʞ j bwxRwx2"±z2yyRyBNqp"B#(b"U\cQ Of;O 68Vr.Q $dv |j(wE~, x5P)+h*X+x |X@lXw( 7P yDH6xwLz89 qt-%.WQ 28zty~;}~T}u 9(X(+(x580w 8=xiTpax(yHxH8xXxxx(h-(-:h.HHHX|zzzOhe4-(-(0HwYox8A# 8x+xx6wzzhzz{zxy x7 8}x8؟ B# xqȠ yx{8zxMCLC+z,7,R+X8xzXҟ XtyP@`x|ybmenTF|Gk`OJ{&uEpUfyz:LgF̭;uEkc7II@IGihle j`#Ye&WVTY]`_ERc7k>V9 5DT?q:Eq6Yy=?{fY=']*I*  O[=[2-R//o; jBWAJr~BiR*D;22|Nٗٗ]*zziuQukij崧يn9hEh 98 %ǒWGe nl5U>o BIK_ȑ&ǑAvG$G( 4q4Jc§c،6{6WcSOu(( r{;]>B]@Z˃\7~oKϩ/}C6׍,d+,ͩE]xnxjnJBcdԴզ"f\@ld 4P*()"ҊKRJRA 7w!j "Ae;UkF'P 9u9G4Pi #4 BF=q;BTVxKE.4%Lc!F K@9# cV`^D$*|bM(M(B2F:\-5(74O74.84EV 99#  !$1 $9t6 6I6| )l9lkmh0nmph0,FO:$ўn:-57U]1k$y( O2%l  y((b6 Mk\g0<LC84 6(Ï,Ӷ,$+U;UMv- )o%sg9/\o.~7s/'0)/bȭ#+:&A(%v&|2#J l)/?5LI yx #M,{7 LG XOF`$y q&^2 p)/V@5S<': :#-ޡ%% %YP3x7. %XLp1Xiov1c/2c-0v3 $ Np3ԝGk|(vF8 V B6:3ӫ 993#FTſ^9[+~(+6.\"T67H<^1%0C-g*$->] *d8 Q&: Z'38:GB@I }  ( ;~)Sm&R$[*k [.%4H7B<,ryS s#=>I+_;  A,B$i NP\pO_$H$ښ'/+vA5B(e2& H&88$)7!##"$F :Vz"3-F; U$"a(bw$֏1         ?@@zD??=>?@>@2@A@AB@?^P/ ABPG@F @@?@C5hCAxD4CzA ⍀PE ⍀PE ⍀PE ⍀PEn ⍀PEU ⍀PvE< ⍀P]E# ⍀PDE k⍀kP+E V⍀VPE A⍀APD ,⍀,PD ⍀PD ⍀PDt ⍀PD[ ⍀P|DB ⍀PcD) ⍀PJD ⍀P1D⍀PDo⍀oPCZ⍀ZPCE⍀EPC0⍀0PCz⍀PCa⍀PCH⍀PiC/⍀PPC⍀P7C⍀PC⍀PC⍀PBs⍀sPB^⍀^PBI⍀IPBg4⍀4PBN⍀PoB5 ⍀ PVB⍀P=B⍀P$B⍀P B⍀PA⍀PA⍀PAw⍀wPAmb⍀bPATM⍀MPuA;8⍀8P\A"#⍀#PCA ⍀P*A⍀PA⍀P@⍀P@⍀P@⍀P@s⍀P@Z{⍀{P{@Af⍀fPb@(Q⍀QPI@<⍀<P0@'⍀'P@⍀P?⍀P?⍀P?⍀P?y⍀P?`⍀P?G⍀Ph?.⍀PO?j⍀jP6?U⍀UP?@⍀@P?+⍀+P>⍀P>⍀P>⍀P>f⍀P>M⍀Pn>4⍀PU>⍀P<>⍀P#>n⍀nP >Y⍀YP=D⍀DP=/⍀/P=⍀P=l⍀P=S⍀Pt=:⍀P[=!⍀PB=⍀P)=⍀P=⍀P<r⍀rP<]⍀]P<H⍀HPUUUUUU? zo ???{Gz?rZ| ?6@yPD#@/7?p= ף?RQ?ףp= ?)\(? h|5@h㈵>X@\(\?j@{Gz?j??0?? ??]9?&?+eG@@ht @Q @9B.@|?5^ @9B.6@iW @@UUUUUU@+eG?&DT!?&DT! x?,4l?B ?Ǝ%>@9B.&@ۦx\T?8dg?e2@Jd?p= ף@G??(\?-"l?UUUUUU?Gz?Q?,b'@333333#@?ףp= ?zG!"@X[#!"@333333?$@?ffffffuYLl>.? 0*?Q?ffffff?@@@)\($@RQ@@= ףp=?Gz?Ů,?II @H}8?8??MbP?& .>)> &FpdWZ      @$\LpD Tp\ U%x x?>>>;>(<p>|=<8><xST ףp= ?RQ?;On?xxd|y0w`vq8k() l) $, ,  t X 4 h  d 5 ' % - - - ` tVSvq -?̯? 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