X<<__text__TEXT>W__data__DATA@| A__cstring__TEXTG|J__literal8__TEXT:d=__picsymbolstub2__TEXT:El=N__la_sym_ptr2__DATA;4>  __nl_symbol_ptr__DATA <>__textcoal_nt__TEXT<> @`d@ P,D$0D$D$@D$ D$:ÐD$$D$D$D$ $:Ð,\$ t$$|$(;T$>D$0T$$9>*L$t$$|$(Y9\$ ,,ÃD$(D$D$ $_9ÐS8T$$5:[Ã<\$0|$8t$4T$H\|$DE$t$94$9\$(tZE<$t$94$9D$\D$(D$D$TD$D$D$@\$ \$$8\$0t$4|$8<É<$붉$뉐<\$0|$8t$4T$H|$D$E$t$ 94$9\$(tZ$E<$t$84$8D$\D$(D$D$TD$D$D$@\$ \$$7\$0t$4|$8<É<$붉$뉐<\$0|$8t$4T$H|$DlD$t$R84$c8\$(tZlD<$t$*84$;8D$\D$(D$D$TD$D$D$@\$ \$$7\$0t$4|$8<É<$붉$뉐<\$0|$8t$4T$H4|$DC$t$74$7\$(tZC<$t$r74$7D$\D$(D$D$TD$D$D$@\$ \$$Y6\$0t$4|$8<É<$붉$뉐<\$0|$8t$4T$H||$DB$t$64$6\$(tZB<$t$64$6D$\D$(D$D$TD$D$D$@\$ \$$5\$0t$4|$8<É<$붉$뉐<\$0|$8t$4T$H|$DDB$t$*64$;6\$(tZDB<$t$64$6D$\D$(D$D$TD$D$D$@\$ \$$4\$0t$4|$8<É<$붉$뉐<\$0|$8t$4T$H |$DtA$t$v54$5\$(tRA<$t$N54$_5D$\D$(D$D$T\$ D$D$@\$$$4\$0t$4|$8<É<$뾉$G2GHH HHHHMHiHHHHHHI I IxI I|IIIIIII;JBJ`JJ!JJKI KKKK LI LnLILLIL%MH@MnMHsMMHMM IM NHN.NH@NiN oNNN oNOoOIOO OPNP VP`PPIPQI@QQIQ@R I`RRRR-SK@SS ISSITFTK`TTKTU U UUHUUcV@VVcV W@XcDV`XnYcwYYHZcV`ZZc$V[[clV[p\H\]I ]^I@^^I^`a'```)a U@aa Iaa Ia&bH@bbIb1cI@ccccccd]dcd3eI@eeIe fI ff Iff If gK ggIg"hK@hhIh7iI`ijI@jjIjjcjkkIkk Il%lH@lvl IllIl"m I@mm ImCn I`nn InWoH[onoHoo oNop p pDp I`ppIpq U qq IqqHrHrc`rr Irr IsAs I`ss It_tHettIttStuuIu*vI@vv Ivv IwTw I`ww>wwsx>xxgyIyz "z@zz K{{K{n|>y||6}>E}`})~ I@~~I~  jHHH EH`ր IH 8I@ ONI`HI e Il8I`I II "z "z!L ̈hKK VP` IljS ˉ!T (@S ˉV  rW y6:@~ IȌH)H@̍ I{h IGh I`FH`HIH lHƑI  Β \ IDh I` IҔIQI`Е Ib I[h IyICI`̙ Iؚ Iݚ _ VP  RҜԜHMI`H@ I IHHH rH I IH H#I@IKIH`HK>H`HGK`K K3R@H R uIH)R@IH  VP@rP ï Iȯ I, I@m II&hI@|IܱI5R@ I& I@lHyHHɶHoHcJI`c Zc ac @c I U I tϼII IPI`I /9@I IoII5I`'I@V I` I U^ I II H;I`Hh"z QI`IR I` I Ig Ih IhIh II II7 p@zU9+ I@ IHGBJ`HfhHHH hH  IL I` Ic I I  Ih I IIh I`H$H@pIzHH2=`H v IHI&1 -@I I# I@ I I+I/KR`!HvII:I@ IIH IKgI II pHHH WIHhHHH @H`I V I[m IHM I` II2 d II  I NI`H Uc\`  IJ I`I I E` IIX_ I) I@ I, I@ IVh I{K IMc`%@nc RK@K I H VP`I VP I YIIl0 q% K S` @HD]H5 I@{ I IEL` UILI` I I$ I@} IIo IHB I`II U I` II  oN< I`  I  I x I r I / @  HI~IIhII "z aH I9H` I U Pc` I5I`I >H`I(K@}K I f k I IjI I6H:KHPm U oN]V I  I3H@ I I@ I I@ Ia  I  % !}! I!! I! " I ""H"#I #f# hU###$$ $$R$$$%I%04 q`%% S%& @&& &&' S@'' '( I (m(I()H )4)H@)`)i))) )))* p@*|*H**I*+BJ +U+h`+,h ,,H,.- I4-H- IN-i- I-- I-'.hH@.. I/[/R ///0 I 00 I00 I021 I@1y1 I11I1C2 `22 23i)@3}3H33H33Q 4r4I44H5#5I@55H56I67I7v8I8a9 I9OO(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 symbolsapprpadicapprpadic(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>=xcenterliftcenterlift(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]chinesechinese(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]classnoclassno(x)=class number of discriminant xclassno2classno2(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[]compositumcompositum(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 discriminantconcatconcat(x,y)=concatenation of x and yconductorGDGDGD1,G,conductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupconductorofcharconductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrconjconj(x)=the algebraic conjugate of xconjvecconjvec(x)=conjugate vector of the algebraic number xcontentcontent(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 moduledegreedegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardenomdenom(x)=denominator of x (or lowest common denominator in case of an array)deplindeplin(x)=finds a linear dependence between the columns of the matrix xderivderiv(x,y)=derivative of x with respect to the main variable of ydetdet(x)=determinant of the matrix xdet2det2(x)=determinant of the matrix x (better for integer entries)detintdetint(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)dirmuldirmul(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-1discdisc(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 conductordiscrayabslistdiscrayabslist(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 conductordivisorsdivisors(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 neigeneigen(x)=eigenvectors of the matrix x given as columns of a matrixeint1eint1(x)=exponential integral E1(x)erfcerfc(x)=complementary error functionetaeta(x)=eta function without the q^(1/24)eulerpeuler=euler()=euler's constant with current precisionevaleval(x)=evaluation of x, replacing variables by their valueexpexp(x)=exponential of xextractextract(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]factmodfactmod(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 tfactorpadicfactorpadic(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-LenstrafactpolGLLfactpol(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)galoisgalois(x)=Galois group of the polynomial x (see manual for group coding)galoisapplygaloisapply(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)gammagamma(x)=gamma function at xgaussgauss(a,b)=gaussian solution of ax=b (a matrix,b vector)gaussmodulogaussmodulo(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 Dgcdgcd(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 SUPPRESSEDhclassnohclassno(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 algorithmhesshess(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 yidealdividealdiv(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 nfidealhermiteidealhermite(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 nfidealinvidealinv(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 differentideallistideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfideallistarchideallistarch(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 nfidealnormidealnorm(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 aidealvalidealval(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)ifif(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 ximage2image2(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)incgam2incgam2(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 polynomialinteginteg(x,y)=formal integration of x with respect to the main variable of yintersectintersect(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 binverseimageinverseimage(x,y)=an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwiseisdiagonalisdiagonal(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 notisprincipalisprincipal(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 obtainedisprincipalrayisprincipalray(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 notisunitisunit(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 otherwisejacobijacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xjbesselhjbesselh(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 SUPPRESSEDkbesselkbessel(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 xkerikeri(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)legendrelegendre(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 otherwisematrixqz2matrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xmatrixqz3matrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsizematsize(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 normmodmod(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 existsmodulargcdmodulargcd(x,y)=gcd of the polynomials x and y using the modular methodmumu(x)=Moebius function of xnewtonpolynewtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnextprimenextprime(x)=smallest prime number>=xnfdetintnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfdivnfdiv(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 smallnfhermitenfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhermitemodnfhermitemod(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 integernfmulnfmul(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 smallnfsmithnfsmith(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 xnumernumer(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 bomegaomega(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-torsionordredordred(x)=reduction of the polynomial x, staying in the same orderpadicprecpadicprec(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 precisionpnqnpnqn(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, yapolredpolred(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 polredpolsympolsym(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 coefficientpolylogLGppolylog(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 xprimeprime(n)=returns the n-th prime (n C-integer)primedecprimedec(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 existsprincipalidealprincipalideal(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 xquadpolyquadpoly(x)=quadratic polynomial corresponding to the discriminant xrandomDGrandom()=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 xreciprecip(x)=reciprocal polynomial of xredimagredimag(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 fregularegula(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 precisionrnfbasisrnfbasis(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*^2rnfequationrnfequation(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 otherwisernfisfreernfisfree(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 notrnflllgramrnflllgram(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 matrixrnfpolredrnfpolred(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*^2rnfsteinitzrnfsteinitz(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 smallrootpadicrootpadic(x,p,r)=p-adic roots of the polynomial x to precision rrootsroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonrootsof1rootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfrootsoldrootsold(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)signiGsign(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))smithcleansmithclean(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 xsrgcdsrgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmsturmsturm(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]subcycloLLDnsubcyclo(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 matrixsylvestermatrixsylvestermatrix(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^)tschirnhaustschirnhaus(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 positiveuntiluntil(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 xvecsortvecsort(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)whilewhile(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 suppressedyPD?k@⍀@PWR+⍀+P>9⍀P% ⍀P ⍀P⍀P⍀P⍀P⍀P⍀Pvqn⍀nP]XY⍀YPD?D⍀DP+::::;;3;L;e;~;;;;$Ë$h@8,GG xG<d\PG<8 GG LGG d\PG<4(G G\|pG\X $GG s`:nj _<:$ < _+@@@ @@@@ @t@p@h@ `@T@P@H@ @@4@0@(@ @@@@ @??? ???? ?????? ?t?p?h? `?T?P?H? @?4?0?(? ???? ?>>> >>>> >>>> >>>>4 >t>p>h> `>T>P>H> @>4>0>(> >>>=== ======== ==== =t=p=h= `=T=P=H= @=4=0=(= =====<<<<<<< <<<< <<<< <t<p<h< `<T<P<H< @<4<0<(<q <<<< <;;; ;;;; ;;;; ;;;; ;t;p;h; `;T;P;H; @;4;0;(; ;;;; ;::::::: :::: :::: :t:p:h: `:T:P:H:@:4:0:(: :::: :999 9999 9999 9999 9t9p9h9 `9T9P9H9 @94909(9 9999 9888 8888 8888 8888 8t8p8h8 `8T8P8H8@84808(8 8888 8777 7777 7777 7777 7t7p7h7 `7T7P7H7 @74707(7 77777666 6666 6666 6666 6t6p6h6 `6T6P6H6 @64606(6 6666 6555 5555 55555555 5t5p5h5 `5T5P5H5 @54505(5 5555 5444 4444 4444 4444 4t4p4h4 `4T4P4H4 @44404(4 4444 4333 3333 3333 3333 3t3p3h3 `3T3P3H3 @34303(3 3333 3222 22222222 2222 2t2p2h2 `2T2P2H2 @24202(2 2222 2111t 1111 1111, 1111+ 1t1p1h1 `1T1P1H1 @14101(1 1111 1000 0000 0000 0000 0t0p0h0 `0T0P0H0 @04000(0 00000/// //// //// //// /t/p/h/ `/T/P/H/ @/4/0/(/ //// /... .... .... .... .t.p.h.} `.T.P.H.| @.4.0.(.{ .... .--- ---- ----z ----w -t-p-h-`-T-P-H-y @-4-0-(-x ----w -,,,v ,,,,u ,,,,s ,,,,q ,t,p,h,o `,T,P,H,S @,4,0,(,n ,,,, ,+++m ++++b ++++l ++++G +t+p+h+j `+T+P+H+h @+4+0+(+g ++++f +***e **** ******c *t*p*h*a `*T*P*H*` @*4*0*(*_ **** *))) ))))m ))))^ ))))l )t)p)h)k `)T)P)H)j @)4)0)()] ))))Z )(((Y ((((X ((((W ((((i (t(p(h(V `(T(P(H(U @(4(0(((T (((( ('''P ''''~ '''' '''' 't'p'h'O `'T'P'H'N @'4'0'('M '''' '&&& &&&&L &&&&K &&&&J &t&p&h&I `&T&P&H&H @&4&0&(&D &&&&C &%%% %%%%* %%%%%%%% %t%p%h% `%T%P%H%o @%4%0%(% %%%% %$$$$$$$B $$$$A $$$$@ $t$p$h$? `$T$P$H$> @$4$0$($= $$$$< $###; ########: ####9 #t#p#h#`#T#P#H#8 @#4#0#(#7 ####6 #"""5 """"4 """"3 """" "t"p"h"2 `"T"P"H" @"4"0"("1 """" "!!!!!!! !!!!!!!!0 !t!p!h!/ `!T!P!H!. @!4!0!(!- !!!!, !   +        *    ) t p h ( ` T P H ' @ 4 0 (       & tph% `TPH$ @40(# " !  d \ tph\ `TPH @40([ [      tph `TPH @40(      tph `TPH @40(      tph`TPH @40(     F tph  `TPH @40(         tph `TPH @40(      tph `TPH @40(      tph `TPH @40(      tph `TPH @40(   E   tph `TPH @40(     tph `TPH@40(n      tph `TPH @40(    tph `TPH @40(      tph `TPH @40( } ~ r tph{ `TPHz @40(y x v u | t tphR `TPHs @40(  Q r  h tphg `TPHf @40( e    /    /    c    b t p h b ` T P H a @ 4 0 ( /    /    `    _    d    ^ t p h ] ` T P H \ @ 4 0 ( [    Z    Y    X    W    V t p h U ` T P H T @ 4 0 ( S    R    Q    w    P    t p h ` T P H @ 4 0 ( O    N    M    L    J    I t p h H ` T P H @ 4 0 ( .    - G F E D tphC `TPHB @40( A @ p ? 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