realprecision = 38 significant digits echo = 1 (on) ? gettime;nfpol=x^5-5*x^3+5*x+25 x^5 - 5*x^3 + 5*x + 25 ? qpol=y^3-y-1;un=Mod(1,qpol);w=Mod(y,qpol);p=un*(x^5-5*x+w) Mod(1, y^3 - y - 1)*x^5 + Mod(-5, y^3 - y - 1)*x + Mod(y, y^3 - y - 1) ? p2=x^5+3021*x^4-786303*x^3-6826636057*x^2-546603588746*x+3853890514072057 x^5 + 3021*x^4 - 786303*x^3 - 6826636057*x^2 - 546603588746*x + 385389051407 2057 ? fa=[11699,6;2392997,2;4987333019653,2] [11699 6] [2392997 2] [4987333019653 2] ? setrand(1);a=matrix(3,5,j,k,vectorv(5,l,random\10^8)); ? setrand(1);as=matrix(3,3,j,k,vectorv(5,l,random\10^8)); ? nf=nfinit(nfpol) [x^5 - 5*x^3 + 5*x + 25, [1, 2], 595125, 45, [[1, -1.08911514572050482502495 27946671612684, -2.4285174907194186068992069565359418365, 0.7194669112891317 8943997506477288225737, -2.5558200350691694950646071159426779972; 1, -0.1383 8372073406036365047976417441696637 - 0.4918163765776864349975328551474152510 7*I, 1.9647119211288133163138753392090569931 + 0.809714924188978951282940822 19556466857*I, -0.072312766896812300380582649294307897074 + 2.19808037538462 76641195195160383234878*I, -0.98796319352507039803950539735452837192 + 1.570 1452385894131769052374806001981109*I; 1, 1.682941293594312776162956161507997 6006 + 2.0500351226010726172974286983598602164*I, -0.75045317576910401286427 186094108607489 + 1.3101462685358123283560773619310445916*I, -0.787420688747 75359433940488309213323154 + 2.1336633893126618034168454610457936018*I, 1.26 58732110596551455718089553258673705 - 2.716479010374315056657802803578983483 5*I], [1, -1.0891151457205048250249527946671612684, -2.428517490719418606899 2069565359418365, 0.71946691128913178943997506477288225737, -2.5558200350691 694950646071159426779972; 1, -0.63020009731174679864801261932183221743, 2.77 44268453177922675968161614046216617, 2.1257676084878153637389368667440155907 , 0.58218204506434277886573208324566973897; 1, 0.353432655843626071347053090 97299828470, 1.1549969969398343650309345170134923246, -2.2703931422814399645 001021653326313849, -2.5581084321144835749447428779547264828; 1, 3.732976416 1953853934603848598678578170, 0.55969309276670831549180550098995851667, 1.34 62427005649082090774405779536603703, -1.450605799314659911085993848253116112 9; 1, -0.36709382900675984113447253685186261580, -2.060599444304916341220349 2228721306665, -2.9210840780604153977562503441379268334, 3.98235222143397020 22296117589048508540], 0, [5, 2, 0, -1, -2; 2, -2, -5, -10, 20; 0, -5, 10, - 10, 5; -1, -10, -10, -17, 1; -2, 20, 5, 1, -8], [345, 0, 200, 110, 177; 0, 3 45, 95, 1, 145; 0, 0, 5, 4, 3; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [63, 3, 0, -6, -9; 3, 8, -5, -1, 16; 0, -5, 22, -10, 0; -6, -1, -10, -14, -9; -9, 16, 0, - 9, -2], [345, [138, 117, 330, 288, -636; -172, -88, 65, 118, -116; 53, 1, 13 8, -173, 65; 1, -172, 54, 191, 106; 0, 118, 173, 225, -34]]], [-2.4285174907 194186068992069565359418365, 1.9647119211288133163138753392090569931 + 0.809 71492418897895128294082219556466857*I, -0.7504531757691040128642718609410860 7489 + 1.3101462685358123283560773619310445916*I], [1, 1/15*x^4 - 2/3*x^2 + 1/3*x + 4/3, x, 2/15*x^4 - 1/3*x^2 + 2/3*x - 1/3, -1/15*x^4 + 1/3*x^3 + 1/3* x^2 - 4/3*x - 2/3], [1, 0, 3, 1, 10; 0, 0, -2, 1, -5; 0, 1, 0, 3, -5; 0, 0, 1, 1, 10; 0, 0, 0, 3, 0], [1, 0, 0, 0, 0, 0, -1, -1, -2, 4, 0, -1, 3, -1, 1, 0, -2, -1, -3, -1, 0, 4, 1, -1, -1; 0, 1, 0, 0, 0, 1, 1, -1, -1, 1, 0, -1, -2, -1, 1, 0, -1, -1, -1, 3, 0, 1, 1, 3, -3; 0, 0, 1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, -2, 0, 1, 0, -1, -1, 0, -1, -2, -1, -1; 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 2, 0, -1; 0, 0, 0, 0, 1, 0, -1, -1 , -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 0, 0, 1, 1, 0, 0, -1]] ? nf1=nfinit(nfpol,2) [x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2, [1, 2], 595125, 4, [[1, -1.089115145 7205048250249527946671612684, 2.4285174907194186068992069565359418365, -0.71 946691128913178943997506477288225734, 2.555820035069169495064607115942677997 1; 1, -0.13838372073406036365047976417441696637 + 0.491816376577686434997532 85514741525107*I, -1.9647119211288133163138753392090569931 + 0.8097149241889 7895128294082219556466856*I, 0.072312766896812300380582649294307897122 + 2.1 980803753846276641195195160383234878*I, 0.9879631935250703980395053973545283 7195 + 1.5701452385894131769052374806001981109*I; 1, 1.682941293594312776162 9561615079976006 + 2.0500351226010726172974286983598602164*I, 0.750453175769 10401286427186094108607490 - 1.3101462685358123283560773619310445915*I, 0.78 742068874775359433940488309213323160 - 2.13366338931266180341684546104579360 16*I, -1.2658732110596551455718089553258673704 + 2.7164790103743150566578028 035789834836*I], [1, -1.0891151457205048250249527946671612684, 2.42851749071 94186068992069565359418365, -0.71946691128913178943997506477288225734, 2.555 8200350691694950646071159426779971; 1, 0.35343265584362607134705309097299828 470, -1.1549969969398343650309345170134923246, 2.270393142281439964500102165 3326313849, 2.5581084321144835749447428779547264828; 1, -0.63020009731174679 864801261932183221744, -2.7744268453177922675968161614046216617, -2.12576760 84878153637389368667440155906, -0.58218204506434277886573208324566973892; 1, 3.7329764161953853934603848598678578170, -0.5596930927667083154918055009899 5851657, -1.3462427005649082090774405779536603700, 1.45060579931465991108599 38482531161132; 1, -0.36709382900675984113447253685186261580, 2.060599444304 9163412203492228721306664, 2.9210840780604153977562503441379268332, -3.98235 22214339702022296117589048508541], 0, [5, 2, 0, 1, 2; 2, -2, 5, 10, -20; 0, 5, 10, -10, 5; 1, 10, -10, -17, 1; 2, -20, 5, 1, -8], [345, 0, 145, 235, 168 ; 0, 345, 250, 344, 200; 0, 0, 5, 4, 3; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [63, 3, 0, 6, 9; 3, 8, 5, 1, -16; 0, 5, 22, -10, 0; 6, 1, -10, -14, -9; 9, -16, 0 , -9, -2], [345, [-138, -117, 330, 288, -636; 172, 88, 65, 118, -116; 53, 1, -138, 173, -65; 1, -172, -54, -191, -106; 0, 118, -173, -225, 34]]], [-1.08 91151457205048250249527946671612684, -0.138383720734060363650479764174416966 37 + 0.49181637657768643499753285514741525107*I, 1.6829412935943127761629561 615079976006 + 2.0500351226010726172974286983598602164*I], [1, x, 1/2*x^4 - 3/2*x^3 + 5/2*x^2 + 2*x - 1, 1/2*x^4 - x^3 + x^2 + 9/2*x + 1, 1/2*x^4 - x^3 + 2*x^2 + 7/2*x + 2], [1, 0, -1, -7, -14; 0, 1, 1, -2, -15; 0, 0, 0, -2, -4; 0, 0, -1, -1, 2; 0, 0, 1, 3, 4], [1, 0, 0, 0, 0, 0, -1, 1, 2, -4, 0, 1, 3, -1, 1, 0, 2, -1, -3, -1, 0, -4, 1, -1, -1; 0, 1, 0, 0, 0, 1, 1, 1, 1, -1, 0, 1, -2, -1, 1, 0, 1, -1, -1, 3, 0, -1, 1, 3, -3; 0, 0, 1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, -1, 2, 1, 1; 0, 0, 0, 1, 0, 0, -1, 0, 0 , 0, 0, 0, -1, -1, -2, 1, 0, -1, 0, 0, 0, 0, -2, 0, 1; 0, 0, 0, 0, 1, 0, 1, -1, -1, 1, 0, -1, 0, -1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 1]] ? nfinit(nfpol,3) [[x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2, [1, 2], 595125, 4, [[1, -1.08911514 57205048250249527946671612684, 2.4285174907194186068992069565359418365, -0.7 1946691128913178943997506477288225734, 2.55582003506916949506460711594267799 71; 1, -0.13838372073406036365047976417441696637 + 0.49181637657768643499753 285514741525107*I, -1.9647119211288133163138753392090569931 + 0.809714924188 97895128294082219556466856*I, 0.072312766896812300380582649294307897122 + 2. 1980803753846276641195195160383234878*I, 0.987963193525070398039505397354528 37195 + 1.5701452385894131769052374806001981109*I; 1, 1.68294129359431277616 29561615079976006 + 2.0500351226010726172974286983598602164*I, 0.75045317576 910401286427186094108607490 - 1.3101462685358123283560773619310445915*I, 0.7 8742068874775359433940488309213323160 - 2.1336633893126618034168454610457936 016*I, -1.2658732110596551455718089553258673704 + 2.716479010374315056657802 8035789834836*I], [1, -1.0891151457205048250249527946671612684, 2.4285174907 194186068992069565359418365, -0.71946691128913178943997506477288225734, 2.55 58200350691694950646071159426779971; 1, 0.3534326558436260713470530909729982 8470, -1.1549969969398343650309345170134923246, 2.27039314228143996450010216 53326313849, 2.5581084321144835749447428779547264828; 1, -0.6302000973117467 9864801261932183221744, -2.7744268453177922675968161614046216617, -2.1257676 084878153637389368667440155906, -0.58218204506434277886573208324566973892; 1 , 3.7329764161953853934603848598678578170, -0.559693092766708315491805500989 95851657, -1.3462427005649082090774405779536603700, 1.4506057993146599110859 938482531161132; 1, -0.36709382900675984113447253685186261580, 2.06059944430 49163412203492228721306664, 2.9210840780604153977562503441379268332, -3.9823 522214339702022296117589048508541], 0, [5, 2, 0, 1, 2; 2, -2, 5, 10, -20; 0, 5, 10, -10, 5; 1, 10, -10, -17, 1; 2, -20, 5, 1, -8], [345, 0, 145, 235, 16 8; 0, 345, 250, 344, 200; 0, 0, 5, 4, 3; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [63, 3, 0, 6, 9; 3, 8, 5, 1, -16; 0, 5, 22, -10, 0; 6, 1, -10, -14, -9; 9, -16, 0, -9, -2], [345, [-138, -117, 330, 288, -636; 172, 88, 65, 118, -116; 53, 1 , -138, 173, -65; 1, -172, -54, -191, -106; 0, 118, -173, -225, 34]]], [-1.0 891151457205048250249527946671612684, -0.13838372073406036365047976417441696 637 + 0.49181637657768643499753285514741525107*I, 1.682941293594312776162956 1615079976006 + 2.0500351226010726172974286983598602164*I], [1, x, 1/2*x^4 - 3/2*x^3 + 5/2*x^2 + 2*x - 1, 1/2*x^4 - x^3 + x^2 + 9/2*x + 1, 1/2*x^4 - x^3 + 2*x^2 + 7/2*x + 2], [1, 0, -1, -7, -14; 0, 1, 1, -2, -15; 0, 0, 0, -2, -4 ; 0, 0, -1, -1, 2; 0, 0, 1, 3, 4], [1, 0, 0, 0, 0, 0, -1, 1, 2, -4, 0, 1, 3, -1, 1, 0, 2, -1, -3, -1, 0, -4, 1, -1, -1; 0, 1, 0, 0, 0, 1, 1, 1, 1, -1, 0 , 1, -2, -1, 1, 0, 1, -1, -1, 3, 0, -1, 1, 3, -3; 0, 0, 1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, -1, 2, 1, 1; 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, -1, -2, 1, 0, -1, 0, 0, 0, 0, -2, 0, 1; 0, 0, 0, 0, 1, 0, 1, -1, -1, 1, 0, -1, 0, -1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 1]], Mod(-1/2*x^4 + 3/2*x^3 - 5/2*x^2 - 2*x + 1, x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2)] ? nfinit(nfpol,4) [x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2, [1, 2], 595125, 4, [[1, -1.089115145 7205048250249527946671612684, 2.4285174907194186068992069565359418365, -0.71 946691128913178943997506477288225734, 2.555820035069169495064607115942677997 1; 1, -0.13838372073406036365047976417441696637 + 0.491816376577686434997532 85514741525107*I, -1.9647119211288133163138753392090569931 + 0.8097149241889 7895128294082219556466856*I, 0.072312766896812300380582649294307897122 + 2.1 980803753846276641195195160383234878*I, 0.9879631935250703980395053973545283 7195 + 1.5701452385894131769052374806001981109*I; 1, 1.682941293594312776162 9561615079976006 + 2.0500351226010726172974286983598602164*I, 0.750453175769 10401286427186094108607490 - 1.3101462685358123283560773619310445915*I, 0.78 742068874775359433940488309213323160 - 2.13366338931266180341684546104579360 16*I, -1.2658732110596551455718089553258673704 + 2.7164790103743150566578028 035789834836*I], [1, -1.0891151457205048250249527946671612684, 2.42851749071 94186068992069565359418365, -0.71946691128913178943997506477288225734, 2.555 8200350691694950646071159426779971; 1, 0.35343265584362607134705309097299828 470, -1.1549969969398343650309345170134923246, 2.270393142281439964500102165 3326313849, 2.5581084321144835749447428779547264828; 1, -0.63020009731174679 864801261932183221744, -2.7744268453177922675968161614046216617, -2.12576760 84878153637389368667440155906, -0.58218204506434277886573208324566973892; 1, 3.7329764161953853934603848598678578170, -0.5596930927667083154918055009899 5851657, -1.3462427005649082090774405779536603700, 1.45060579931465991108599 38482531161132; 1, -0.36709382900675984113447253685186261580, 2.060599444304 9163412203492228721306664, 2.9210840780604153977562503441379268332, -3.98235 22214339702022296117589048508541], 0, [5, 2, 0, 1, 2; 2, -2, 5, 10, -20; 0, 5, 10, -10, 5; 1, 10, -10, -17, 1; 2, -20, 5, 1, -8], [345, 0, 145, 235, 168 ; 0, 345, 250, 344, 200; 0, 0, 5, 4, 3; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [63, 3, 0, 6, 9; 3, 8, 5, 1, -16; 0, 5, 22, -10, 0; 6, 1, -10, -14, -9; 9, -16, 0 , -9, -2], [345, [-138, -117, 330, 288, -636; 172, 88, 65, 118, -116; 53, 1, -138, 173, -65; 1, -172, -54, -191, -106; 0, 118, -173, -225, 34]]], [-1.08 91151457205048250249527946671612684, -0.138383720734060363650479764174416966 37 + 0.49181637657768643499753285514741525107*I, 1.6829412935943127761629561 615079976006 + 2.0500351226010726172974286983598602164*I], [1, x, 1/2*x^4 - 3/2*x^3 + 5/2*x^2 + 2*x - 1, 1/2*x^4 - x^3 + x^2 + 9/2*x + 1, 1/2*x^4 - x^3 + 2*x^2 + 7/2*x + 2], [1, 0, -1, -7, -14; 0, 1, 1, -2, -15; 0, 0, 0, -2, -4; 0, 0, -1, -1, 2; 0, 0, 1, 3, 4], [1, 0, 0, 0, 0, 0, -1, 1, 2, -4, 0, 1, 3, -1, 1, 0, 2, -1, -3, -1, 0, -4, 1, -1, -1; 0, 1, 0, 0, 0, 1, 1, 1, 1, -1, 0, 1, -2, -1, 1, 0, 1, -1, -1, 3, 0, -1, 1, 3, -3; 0, 0, 1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, -1, 2, 1, 1; 0, 0, 0, 1, 0, 0, -1, 0, 0 , 0, 0, 0, -1, -1, -2, 1, 0, -1, 0, 0, 0, 0, -2, 0, 1; 0, 0, 0, 0, 1, 0, 1, -1, -1, 1, 0, -1, 0, -1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 1]] ? nf3=nfinit(x^6+108); ? nf4=nfinit(x^3-10*x+8) [x^3 - 10*x + 8, [3, 0], 568, 2, [[1, -0.36332823793268357037416860931988791 957, -3.1413361156553641347759399165844441384; 1, -1.76155718183189058754537 11274124874988, 2.6261980685272936133764995500786243868; 1, 3.12488541976457 41579195397367323754184, -0.48486195287192947860055963349418024847], [1, -0. 36332823793268357037416860931988791957, -3.141336115655364134775939916584444 1384; 1, -1.7615571818318905875453711274124874988, 2.62619806852729361337649 95500786243868; 1, 3.1248854197645741579195397367323754184, -0.4848619528719 2947860055963349418024847], 0, [3, 1, -1; 1, 13, -5; -1, -5, 17], [284, 76, 46; 0, 2, 0; 0, 0, 1], [98, -6, 4; -6, 25, 7; 4, 7, 19], [284, [60, 418, -20 4; 105, 270, -2; 1, 104, -46]]], [-3.5046643535880477051501085259043320579, 0.86464088669540302583112842266613688800, 2.64002346689264467931898010323819 51699], [1, 1/2*x^2 + x - 3, -1/2*x^2 + 3], [1, 0, 6; 0, 1, 0; 0, 1, -2], [1 , 0, 0, 0, 4, -2, 0, -2, 6; 0, 1, 0, 1, 2, 0, 0, 0, -2; 0, 0, 1, 0, 1, -1, 1 , -1, -1]] ? setrand(1);bnf2=bnfinit(qpol);nf2=bnf2[7]; ? setrand(1);bnf=bnfinit(x^2-x-57,,[0.2,0.2]) [Mat(3), Mat([1, 2, 1, 2, 1, 2, 1, 2, 1]), [-2.71246530518434397468087951060 61300699 + 3.1415926535897932384626433832795028842*I; 2.71246530518434397468 08795106061300699 + 6.2831853071795864769252867665590057684*I], [1.790341756 6977293763292119206302198761, 1.2897619530652735025030086072395031017 + 3.14 15926535897932384626433832795028842*I, 0.70148550268542821846861610071436900 870, 0.E-38, 0.50057980363245587382620331339071677437 + 3.141592653589793238 4626433832795028842*I, 1.0888562540123011578605958199158508674, 1.7241634548 149836441438434283070556827 + 3.1415926535897932384626433832795028842*I, 2.3 691776609073168802909916438727108512 + 3.14159265358979323846264338327950288 42*I, -0.57883590420958750396177972324249097505 + 3.141592653589793238462643 3832795028842*I, 0.066178301882745732185368492323164193433 + 3.1415926535897 932384626433832795028842*I; -1.7903417566977293763292119206302198761, -1.289 7619530652735025030086072395031018, -0.7014855026854282184686161007143690087 0, 0.E-38, -0.50057980363245587382620331339071677436, -1.0888562540123011578 605958199158508674, -1.7241634548149836441438434283070556827, -2.36917766090 73168802909916438727108512 + 3.1415926535897932384626433832795028842*I, 0.57 883590420958750396177972324249097505 + 3.14159265358979323846264338327950288 42*I, -0.066178301882745732185368492323164193433], [[3, [0, 1]~, 1, 1, [1, 5 7; 1, 0]], [5, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [11, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [3, [1, 1]~, 1, 1, [0, 57; 1, -1]], [5, [2, 1]~, 1, 1, [-1, 57; 1, - 2]], [11, [2, 1]~, 1, 1, [-1, 57; 1, -2]], [19, [1, 1]~, 1, 1, [0, 57; 1, -1 ]], [17, [3, 1]~, 1, 1, [-2, 57; 1, -3]], [17, [-2, 1]~, 1, 1, [3, 57; 1, 2] ], [19, [0, 1]~, 1, 1, [1, 57; 1, 0]]], 0, [x^2 - x - 57, [2, 0], 229, 1, [[ 1, -8.0663729752107779635959310246705326058; 1, 7.06637297521077796359593102 46705326058], [1, -8.0663729752107779635959310246705326058; 1, 7.06637297521 07779635959310246705326058], 0, [2, -1; -1, 115], [229, 115; 0, 1], [115, 1; 1, 2], [229, [115, 57; 1, 114]]], [-7.0663729752107779635959310246705326058 , 8.0663729752107779635959310246705326058], [1, x - 1], [1, 1; 0, 1], [1, 0, 0, 57; 0, 1, 1, -1]], [[3, [3], [[3, 0; 0, 1]]], 2.712465305184343974680879 5106061300699, 1, [2, -1], [x + 7]], [Mat(1), [[0, 0]], [[1.7903417566977293 763292119206302198761, -1.7903417566977293763292119206302198761]]], 0] ? setrand(1);bnfinit(x^2-x-100000,1) [Mat(5), Mat([3, 2, 1, 2, 0, 2, 3, 3, 2, 0, 0, 4, 1, 2, 3, 3, 2, 3]), [-129. 82045011403975460991182396195022419 + 6.283185307179586476925286766559005768 3*I; 129.82045011403975460991182396195022419 + 6.283185307179586476925286766 5590057683*I], [-41.811264589129943393339502258694361489 + 3.141592653589793 2384626433832795028842*I, 9.2399004147902289816376260438840931575 + 5.934729 84 E-67*I, -11.874609881075406725097315997431161032 + 6.28318530717958647692 52867665590057683*I, -389.46135034211926382973547188585067257 + 6.2831853071 795864769252867665590057683*I, 78.769285110119582234934695659371769544 + 3.1 415926535897932384626433832795028842*I, -843.8329257412584049644268557526764 5724 + 6.2831853071795864769252867665590057683*I, 239.9341511615634437073347 4008902230144 + 3.1415926535897932384626433832795028842*I, 619.1655958621442 3638131693354848792049 + 3.1415926535897932384626433832795028842*I, 1003.490 0875014837815467353253879620702 + 3.1415926535897932384626433832795028842*I, -266.37865140637519728060388387495508649 + 6.283185307179586476925286766559 0057683*I, 177.48876918560798860724474244465791207 + 3.141592653589793238462 6433832795028842*I, -177.48876918560798860724474244465791207 + 3.14159265358 97932384626433832795028842*I, 574.16020455447134808446931672347404239 + 4.74 7783872 E-65*I, -486.15101902956153686789699502021817969 + 6.283185307179586 4769252867665590057683*I, -26.831076484481330319708743069401142309 + 3.14159 26535897932384626433832795028842*I, -144.80063821868836768354258315124344337 + 6.2831853071795864769252867665590057683*I, 367.35683481950538594888487746 203445803 + 9.49556774 E-66*I, 782.79899433071641016621942625073685276 + 3.1 415926535897932384626433832795028842*I, -824.6102589198463535595589285094312 1425 + 6.2831853071795864769252867665590057683*I; 41.81126458912994339333950 2258694361489 + 3.1415926535897932384626433832795028842*I, -9.23990041479022 89816376260438840931575 + 3.1415926535897932384626433832795028842*I, 11.8746 09881075406725097315997431161032 + 3.1415926535897932384626433832795028842*I , 389.46135034211926382973547188585067257 + 6.283185307179586476925286766559 0057683*I, -78.769285110119582234934695659371769544 + 6.28318530717958647692 52867665590057683*I, 843.83292574125840496442685575267645724 + 3.14159265358 97932384626433832795028842*I, -239.93415116156344370733474008902230144 + 4.7 4778387 E-66*I, -619.16559586214423638131693354848792049 + 1.899113549 E-65* I, -1003.4900875014837815467353253879620702 + 3.798227099 E-65*I, 266.378651 40637519728060388387495508649 + 3.1415926535897932384626433832795028842*I, - 177.48876918560798860724474244465791207 + 3.14159265358979323846264338327950 28842*I, 177.48876918560798860724474244465791207 + 3.14159265358979323846264 33832795028842*I, -574.16020455447134808446931672347404239 + 2.848670324 E-6 5*I, 486.15101902956153686789699502021817969 + 6.283185307179586476925286766 5590057683*I, 26.831076484481330319708743069401142309 + 1.780418952 E-66*I, 144.80063821868836768354258315124344337 + 3.14159265358979323846264338327950 28842*I, -367.35683481950538594888487746203445803 + 3.1415926535897932384626 433832795028842*I, -782.79899433071641016621942625073685276 + 3.141592653589 7932384626433832795028842*I, 824.61025891984635355955892850943121425 + 6.283 1853071795864769252867665590057683*I], [[2, [2, 1]~, 1, 1, [1, 100000; 1, 0] ], [5, [5, 1]~, 1, 1, [1, 100000; 1, 0]], [13, [-5, 1]~, 1, 1, [6, 100000; 1 , 5]], [2, [3, 1]~, 1, 1, [0, 100000; 1, -1]], [5, [6, 1]~, 1, 1, [0, 100000 ; 1, -1]], [7, [4, 1]~, 2, 1, [-3, 100000; 1, -4]], [41, [7, 1]~, 1, 1, [-6, 100000; 1, -7]], [13, [6, 1]~, 1, 1, [-5, 100000; 1, -6]], [17, [15, 1]~, 1 , 1, [3, 100000; 1, 2]], [17, [20, 1]~, 1, 1, [-2, 100000; 1, -3]], [23, [-6 , 1]~, 1, 1, [7, 100000; 1, 6]], [23, [7, 1]~, 1, 1, [-6, 100000; 1, -7]], [ 29, [-13, 1]~, 1, 1, [14, 100000; 1, 13]], [29, [14, 1]~, 1, 1, [-13, 100000 ; 1, -14]], [31, [24, 1]~, 1, 1, [8, 100000; 1, 7]], [31, [39, 1]~, 1, 1, [- 7, 100000; 1, -8]], [41, [-6, 1]~, 1, 1, [7, 100000; 1, 6]], [43, [-15, 1]~, 1, 1, [16, 100000; 1, 15]], [43, [16, 1]~, 1, 1, [-15, 100000; 1, -16]]], 0 , [x^2 - x - 100000, [2, 0], 400001, 1, [[1, -316.72816130129840161392089489 603747004; 1, 315.72816130129840161392089489603747004], [1, -316.72816130129 840161392089489603747004; 1, 315.72816130129840161392089489603747004], 0, [2 , -1; -1, 200001], [400001, 200001; 0, 1], [200001, 1; 1, 2], [400001, [2000 01, 100000; 1, 200000]]], [-315.72816130129840161392089489603747004, 316.728 16130129840161392089489603747004], [1, x - 1], [1, 1; 0, 1], [1, 0, 0, 10000 0; 0, 1, 1, -1]], [[5, [5], [[2, 0; 0, 1]]], 129.820450114039754609911823961 95022419, 1, [2, -1], [37955488401901378100630325489636915406833608260923833 6*x + 119836165644250789990462835950022871665178127611316131167]], [Mat(1), [[0, 0]], [[-41.811264589129943393339502258694361489 + 3.1415926535897932384 626433832795028842*I, 41.811264589129943393339502258694361489 + 3.1415926535 897932384626433832795028842*I]]], 0] ? \p19 realprecision = 19 significant digits ? setrand(1);sbnf=bnfinit(x^3-x^2-14*x-1,3) [x^3 - x^2 - 14*x - 1, 3, 10889, [1, x, x^2 - x - 9], [-3.233732695981516673 , -0.07182350902743636344, 4.305556205008953036], [10889, 5698, 8994; 0, 1, 0; 0, 0, 1], Mat(2), Mat([1, 1, 0, 1, 0, 1, 1, 1]), [9, 15, 16, 33, 39, 17, 10, 57, 69], [2, [-1, 0, 0]~], [[0, 1, 0]~, [5, 3, 1]~], [[-4, -1, 2, 1, 10, 3, 3, 7, 2; 1, 1, 1, 1, 5, 1, 0, 2, 1; 0, 0, 0, 0, 1, 0, 0, 0, -1], 0]] ? \p38 realprecision = 38 significant digits ? bnrinit(bnf,[[5,4;0,1],[1,0]],1) [[Mat(3), Mat([1, 2, 1, 2, 1, 2, 1, 2, 1]), [-2.7124653051843439746808795106 061300699 + 3.1415926535897932384626433832795028842*I; 2.7124653051843439746 808795106061300699 + 6.2831853071795864769252867665590057684*I], [1.79034175 66977293763292119206302198761, 1.2897619530652735025030086072395031017 + 3.1 415926535897932384626433832795028842*I, 0.7014855026854282184686161007143690 0870, 0.E-38, 0.50057980363245587382620331339071677437 + 3.14159265358979323 84626433832795028842*I, 1.0888562540123011578605958199158508674, 1.724163454 8149836441438434283070556827 + 3.1415926535897932384626433832795028842*I, 2. 3691776609073168802909916438727108512 + 3.1415926535897932384626433832795028 842*I, -0.57883590420958750396177972324249097505 + 3.14159265358979323846264 33832795028842*I, 0.066178301882745732185368492323164193433 + 3.141592653589 7932384626433832795028842*I; -1.7903417566977293763292119206302198761, -1.28 97619530652735025030086072395031018, -0.701485502685428218468616100714369008 70, 0.E-38, -0.50057980363245587382620331339071677436, -1.088856254012301157 8605958199158508674, -1.7241634548149836441438434283070556827, -2.3691776609 073168802909916438727108512 + 3.1415926535897932384626433832795028842*I, 0.5 7883590420958750396177972324249097505 + 3.1415926535897932384626433832795028 842*I, -0.066178301882745732185368492323164193433], [[3, [0, 1]~, 1, 1, [1, 57; 1, 0]], [5, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [11, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [3, [1, 1]~, 1, 1, [0, 57; 1, -1]], [5, [2, 1]~, 1, 1, [-1, 57; 1, -2]], [11, [2, 1]~, 1, 1, [-1, 57; 1, -2]], [19, [1, 1]~, 1, 1, [0, 57; 1, - 1]], [17, [3, 1]~, 1, 1, [-2, 57; 1, -3]], [17, [-2, 1]~, 1, 1, [3, 57; 1, 2 ]], [19, [0, 1]~, 1, 1, [1, 57; 1, 0]]], 0, [x^2 - x - 57, [2, 0], 229, 1, [ [1, -8.0663729752107779635959310246705326058; 1, 7.0663729752107779635959310 246705326058], [1, -8.0663729752107779635959310246705326058; 1, 7.0663729752 107779635959310246705326058], 0, [2, -1; -1, 115], [229, 115; 0, 1], [115, 1 ; 1, 2], [229, [115, 57; 1, 114]]], [-7.066372975210777963595931024670532605 8, 8.0663729752107779635959310246705326058], [1, x - 1], [1, 1; 0, 1], [1, 0 , 0, 57; 0, 1, 1, -1]], [[3, [3], [[3, 0; 0, 1]]], 2.71246530518434397468087 95106061300699, 1, [2, -1], [x + 7]], [Mat(1), [[0, 0]], [[1.790341756697729 3763292119206302198761, -1.7903417566977293763292119206302198761]]], [0, [Ma t([[6, 1]~, 1])]]], [[[5, 4; 0, 1], [1, 0]], [8, [4, 2], [[2, 0]~, [-4, 0]~] ], Mat([[5, [-1, 1]~, 1, 1, [2, 1]~], 1]), [[[[4], [[2, 0]~], [[2, 0]~], [[0 ]~], 1]], [[2], [-4], Mat(1)]], [1, 0; 0, 1]], [1], Mat([1, -3, -6]), [12, [ 12], [[3, 0; 0, 1]]], [[0, 0; 0, -1], [1, -1; 1, 1]]] ? bnr=bnrclass(bnf,[[5,4;0,1],[1,0]],2) [[Mat(3), Mat([1, 2, 1, 2, 1, 2, 1, 2, 1]), [-2.7124653051843439746808795106 061300699 + 3.1415926535897932384626433832795028842*I; 2.7124653051843439746 808795106061300699 + 6.2831853071795864769252867665590057684*I], [1.79034175 66977293763292119206302198761, 1.2897619530652735025030086072395031017 + 3.1 415926535897932384626433832795028842*I, 0.7014855026854282184686161007143690 0870, 0.E-38, 0.50057980363245587382620331339071677437 + 3.14159265358979323 84626433832795028842*I, 1.0888562540123011578605958199158508674, 1.724163454 8149836441438434283070556827 + 3.1415926535897932384626433832795028842*I, 2. 3691776609073168802909916438727108512 + 3.1415926535897932384626433832795028 842*I, -0.57883590420958750396177972324249097505 + 3.14159265358979323846264 33832795028842*I, 0.066178301882745732185368492323164193433 + 3.141592653589 7932384626433832795028842*I; -1.7903417566977293763292119206302198761, -1.28 97619530652735025030086072395031018, -0.701485502685428218468616100714369008 70, 0.E-38, -0.50057980363245587382620331339071677436, -1.088856254012301157 8605958199158508674, -1.7241634548149836441438434283070556827, -2.3691776609 073168802909916438727108512 + 3.1415926535897932384626433832795028842*I, 0.5 7883590420958750396177972324249097505 + 3.1415926535897932384626433832795028 842*I, -0.066178301882745732185368492323164193433], [[3, [0, 1]~, 1, 1, [1, 57; 1, 0]], [5, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [11, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [3, [1, 1]~, 1, 1, [0, 57; 1, -1]], [5, [2, 1]~, 1, 1, [-1, 57; 1, -2]], [11, [2, 1]~, 1, 1, [-1, 57; 1, -2]], [19, [1, 1]~, 1, 1, [0, 57; 1, - 1]], [17, [3, 1]~, 1, 1, [-2, 57; 1, -3]], [17, [-2, 1]~, 1, 1, [3, 57; 1, 2 ]], [19, [0, 1]~, 1, 1, [1, 57; 1, 0]]], 0, [x^2 - x - 57, [2, 0], 229, 1, [ [1, -8.0663729752107779635959310246705326058; 1, 7.0663729752107779635959310 246705326058], [1, -8.0663729752107779635959310246705326058; 1, 7.0663729752 107779635959310246705326058], 0, [2, -1; -1, 115], [229, 115; 0, 1], [115, 1 ; 1, 2], [229, [115, 57; 1, 114]]], [-7.066372975210777963595931024670532605 8, 8.0663729752107779635959310246705326058], [1, x - 1], [1, 1; 0, 1], [1, 0 , 0, 57; 0, 1, 1, -1]], [[3, [3], [[3, 0; 0, 1]]], 2.71246530518434397468087 95106061300699, 1, [2, -1], [x + 7]], [Mat(1), [[0, 0]], [[1.790341756697729 3763292119206302198761, -1.7903417566977293763292119206302198761]]], [0, [Ma t([[6, 1]~, 1])]]], [[[5, 4; 0, 1], [1, 0]], [8, [4, 2], [[2, 0]~, [-4, 0]~] ], Mat([[5, [-1, 1]~, 1, 1, [2, 1]~], 1]), [[[[4], [[2, 0]~], [[2, 0]~], [[0 ]~], 1]], [[2], [-4], Mat(1)]], [1, 0; 0, 1]], [1], Mat([1, -3, -6]), [12, [ 12], [[3, 0; 0, 1]]], [[0, 0; 0, -1], [1, -1; 1, 1]]] ? rnfinit(nf2,x^5-x-2) [x^5 - x - 2, [], [[49744, 0, 0; 0, 49744, 0; 0, 0, 49744], [3109, 0, 0]~], 1, [], [], [[1, x, x^2, x^3, x^4], [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0; 0 , 1, 0; 0, 0, 1], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0; 0, 1, 0; 0, 0, 1]]], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [], [y^3 - y - 1, [1, 1], -23, 1, [[1, 0.754 87766624669276004950889635852869190, 1.3247179572447460259609088544780973407 ; 1, -0.87743883312334638002475444817926434595 - 0.7448617666197442365931704 2860439236724*I, -0.66235897862237301298045442723904867036 + 0.5622795120623 0124389918214490937306149*I], [1, 0.75487766624669276004950889635852869190, 1.3247179572447460259609088544780973407; 1, -1.62230059974309061661792487678 36567132, -0.10007946656007176908127228232967560887; 1, -0.13257706650360214 343158401957487197871, -1.2246384906846742568796365721484217319], 0, [3, -1, 0; -1, 1, 3; 0, 3, 2], [23, 16, 13; 0, 1, 0; 0, 0, 1], [7, -2, 3; -2, -6, 9 ; 3, 9, -2], [23, [10, 1, 8; 7, 3, 1; 1, 7, 10]]], [1.3247179572447460259609 088544780973407, -0.66235897862237301298045442723904867036 + 0.5622795120623 0124389918214490937306149*I], [1, y^2 - 1, y], [1, 0, 1; 0, 0, 1; 0, 1, 0], [1, 0, 0, 0, 0, 1, 0, 1, 1; 0, 1, 0, 1, -1, 0, 0, 0, 1; 0, 0, 1, 0, 1, 0, 1, 0, 0]], [x^15 - 5*x^13 + 5*x^12 + 7*x^11 - 26*x^10 - 5*x^9 + 45*x^8 + 158*x ^7 - 98*x^6 + 110*x^5 - 190*x^4 + 189*x^3 + 144*x^2 + 25*x + 1, Mod(39516536 165538345/83718587879473471*x^14 - 6500512476832995/83718587879473471*x^13 - 196215472046117185/83718587879473471*x^12 + 229902227480108910/837185878794 73471*x^11 + 237380704030959181/83718587879473471*x^10 - 1064931988160773805 /83718587879473471*x^9 - 20657086671714300/83718587879473471*x^8 + 177288520 5999206010/83718587879473471*x^7 + 5952033217241102348/83718587879473471*x^6 - 4838840187320655696/83718587879473471*x^5 + 5180390720553188700/837185878 79473471*x^4 - 8374015687535120430/83718587879473471*x^3 + 89077447279150402 21/83718587879473471*x^2 + 4155976664123434381/83718587879473471*x + 3189202 15718580450/83718587879473471, x^15 - 5*x^13 + 5*x^12 + 7*x^11 - 26*x^10 - 5 *x^9 + 45*x^8 + 158*x^7 - 98*x^6 + 110*x^5 - 190*x^4 + 189*x^3 + 144*x^2 + 2 5*x + 1), -1], 0] ? bnfcertify(bnf) 1 ? setrand(1);bnfclassunit(x^4-7,2,[0.2,0.2]) [x^4 - 7] [[2, 1]] [[-87808, 1]] [[1, x, x^2, x^3]] [[2, [2], [[3, 2, 2, 2; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]]] [14.229975145405511722395637833443108790] [1] ? setrand(1);bnfclassunit(x^2-x-100000) *** bnfclassunit: Warning: insufficient precision for fundamental units, not given. [x^2 - x - 100000] [[2, 0]] [[400001, 1]] [[1, x - 1]] [[5, [5], [[43, 28; 0, 1]]]] [129.82045011403975460991182396195022419] [1] [[2, -1]] [[;]] ? setrand(1);bnfclassunit(x^2-x-100000,1) [x^2 - x - 100000] [[2, 0]] [[400001, 1]] [[1, x - 1]] [[5, [5], [[2, 0; 0, 1]]]] [129.82045011403975460991182396195022419] [1] [[2, -1]] [[379554884019013781006303254896369154068336082609238336*x + 119836165644250 789990462835950022871665178127611316131167]] ? setrand(1);bnfclassunit(x^4+24*x^2+585*x+1791,,[0.1,0.1]) [x^4 + 24*x^2 + 585*x + 1791] [[0, 2]] [[18981, 3087]] [[1, -10/1029*x^3 + 13/343*x^2 - 165/343*x - 1478/343, 17/1029*x^3 - 32/1029 *x^2 + 109/343*x + 2444/343, -43/1029*x^3 + 202/1029*x^2 - 538/343*x - 5395/ 343]] [[4, [4], [[7, 3, 0, 0; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]]] [3.7941269688216589341408274220859400303] [1] [[6, -10/1029*x^3 + 13/343*x^2 - 165/343*x - 1135/343]] [[4/1029*x^3 + 53/1029*x^2 + 66/343*x + 111/343]] ? setrand(1);bnfclgp(17) [1, [], []] ? setrand(1);bnfclgp(-31) [3, [3], [Qfb(2, 1, 4)]] ? setrand(1);bnfclgp(x^4+24*x^2+585*x+1791) [4, [4], [[7, 3, 0, 0; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]] ? bnrconductor(bnf,[[25,14;0,1],[1,1]]) [[5, 4; 0, 1], [1, 0]] ? bnrconductorofchar(bnr,[2]) [[5, 4; 0, 1], [0, 0]] ? bnfisprincipal(bnf,[5,2;0,1],0) [1]~ ? bnfisprincipal(bnf,[5,2;0,1]) [[1]~, [7/3, 1/3]~] ? bnfisunit(bnf,Mod(3405*x-27466,x^2-x-57)) [-4, Mod(1, 2)]~ ? \p19 realprecision = 19 significant digits ? bnfmake(sbnf) [Mat(2), Mat([1, 1, 0, 1, 0, 1, 1, 1]), [1.173637103435061715 + 3.1415926535 89793238*I, -4.562279014988837911 + 3.141592653589793238*I; -2.6335434327389 76050 + 3.141592653589793238*I, 1.420330600779487358 + 3.141592653589793238* I; 1.459906329303914335, 3.141948414209350543], [1.246346989334819161 + 3.14 1592653589793238*I, 0.5404006376129469728 + 3.141592653589793238*I, -0.69263 91142471042844 + 3.141592653589793238*I, 0.004375616572659815433 + 3.1415926 53589793238*I, -0.8305625946607188641 + 3.141592653589793238*I, -1.990056445 584799713 + 3.141592653589793238*I, 0.E-38, -1.977791147836553954, 0.3677262 014027817706 + 3.141592653589793238*I; 0.6716827432867392934 + 3.14159265358 9793238*I, -0.8333219883742404172 + 3.141592653589793238*I, -0.2461086674077 943078, -0.8738318043071131265, -1.552661549868775854, 0.5379005671092853266 , 0.E-38, 0.5774919091398324093, 0.9729063188316092381; -1.91802973262155845 4, 0.2929213507612934445, 0.9387477816548985924, 0.8694561877344533111, 2.38 3224144529494718, 1.452155878475514387, 0.E-38, 1.400299238696721545, -1.340 632520234391008], [[3, [-1, 1, 0]~, 1, 1, [1, 1, 1]~], [5, [-1, 1, 0]~, 1, 1 , [0, 1, 1]~], [5, [2, 1, 0]~, 1, 1, [1, -2, 1]~], [11, [1, 1, 0]~, 1, 1, [- 3, -1, 1]~], [13, [19, 1, 0]~, 1, 1, [-2, -6, 1]~], [5, [3, 1, 0]~, 1, 1, [2 , 2, 1]~], [3, [10, 1, 1]~, 1, 2, [-1, 1, 0]~], [19, [-6, 1, 0]~, 1, 1, [6, 6, 1]~], [23, [-10, 1, 0]~, 1, 1, [-7, 10, 1]~]]~, 0, [x^3 - x^2 - 14*x - 1, [3, 0], 10889, 1, [[1, -3.233732695981516673, 4.690759845041404812; 1, -0.0 7182350902743636345, -8.923017874523549404; 1, 4.305556205008953036, 5.23225 8029482144592], [1, -3.233732695981516673, 4.690759845041404812; 1, -0.07182 350902743636345, -8.923017874523549404; 1, 4.305556205008953036, 5.232258029 482144592], 0, [3, 1, 1; 1, 29, 8; 1, 8, 129], [10889, 5698, 8994; 0, 1, 0; 0, 0, 1], [3677, -121, -21; -121, 386, -23; -21, -23, 86], [10889, [1899, 46 720, 5235; 5191, 7095, 25956; 1, 5191, 1895]]], [-3.233732695981516673, -0.0 7182350902743636345, 4.305556205008953036], [1, x, x^2 - x - 9], [1, 0, 9; 0 , 1, 1; 0, 0, 1], [1, 0, 0, 0, 9, 1, 0, 1, 44; 0, 1, 0, 1, 1, 5, 0, 5, 1; 0, 0, 1, 0, 1, 0, 1, 0, -4]], [[2, [2], [[3, 2, 0; 0, 1, 0; 0, 0, 1]]], 10.348 00724602768001, 1, [2, -1], [x, x^2 + 2*x - 4]], [Mat(1), [[0.E-38, 0.E-38, 0.E-38]], [[1.246346989334819161 + 3.141592653589793238*I, 0.671682743286739 2934 + 3.141592653589793238*I, -1.918029732621558454]]], [[-4, -1, 2, 1, 10, 3, 3, 7, 2; 1, 1, 1, 1, 5, 1, 0, 2, 1; 0, 0, 0, 0, 1, 0, 0, 0, -1], 0]] ? \p38 realprecision = 38 significant digits ? bnfnarrow(bnf) [3, [3], [[3, 0; 0, 1]]] ? bnfreg(x^2-x-57) 2.7124653051843439746808795106061300699 ? bnfsignunit(bnf) [-1] [1] ? bnfunit(bnf) [x + 7] ? bnrclass(bnf,[[5,4;0,1],[1,0]]) [12, [12], [[3, 0; 0, 1]]] ? bnr2=bnrclass(bnf,[[25,14;0,1],[1,1]],2) [[Mat(3), Mat([1, 2, 1, 2, 1, 2, 1, 2, 1]), [-2.7124653051843439746808795106 061300699 + 3.1415926535897932384626433832795028842*I; 2.7124653051843439746 808795106061300699 + 6.2831853071795864769252867665590057684*I], [1.79034175 66977293763292119206302198761, 1.2897619530652735025030086072395031017 + 3.1 415926535897932384626433832795028842*I, 0.7014855026854282184686161007143690 0870, 0.E-38, 0.50057980363245587382620331339071677437 + 3.14159265358979323 84626433832795028842*I, 1.0888562540123011578605958199158508674, 1.724163454 8149836441438434283070556827 + 3.1415926535897932384626433832795028842*I, 2. 3691776609073168802909916438727108512 + 3.1415926535897932384626433832795028 842*I, -0.57883590420958750396177972324249097505 + 3.14159265358979323846264 33832795028842*I, 0.066178301882745732185368492323164193433 + 3.141592653589 7932384626433832795028842*I; -1.7903417566977293763292119206302198761, -1.28 97619530652735025030086072395031018, -0.701485502685428218468616100714369008 70, 0.E-38, -0.50057980363245587382620331339071677436, -1.088856254012301157 8605958199158508674, -1.7241634548149836441438434283070556827, -2.3691776609 073168802909916438727108512 + 3.1415926535897932384626433832795028842*I, 0.5 7883590420958750396177972324249097505 + 3.1415926535897932384626433832795028 842*I, -0.066178301882745732185368492323164193433], [[3, [0, 1]~, 1, 1, [1, 57; 1, 0]], [5, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [11, [-1, 1]~, 1, 1, [2, 57; 1, 1]], [3, [1, 1]~, 1, 1, [0, 57; 1, -1]], [5, [2, 1]~, 1, 1, [-1, 57; 1, -2]], [11, [2, 1]~, 1, 1, [-1, 57; 1, -2]], [19, [1, 1]~, 1, 1, [0, 57; 1, - 1]], [17, [3, 1]~, 1, 1, [-2, 57; 1, -3]], [17, [-2, 1]~, 1, 1, [3, 57; 1, 2 ]], [19, [0, 1]~, 1, 1, [1, 57; 1, 0]]], 0, [x^2 - x - 57, [2, 0], 229, 1, [ [1, -8.0663729752107779635959310246705326058; 1, 7.0663729752107779635959310 246705326058], [1, -8.0663729752107779635959310246705326058; 1, 7.0663729752 107779635959310246705326058], 0, [2, -1; -1, 115], [229, 115; 0, 1], [115, 1 ; 1, 2], [229, [115, 57; 1, 114]]], [-7.066372975210777963595931024670532605 8, 8.0663729752107779635959310246705326058], [1, x - 1], [1, 1; 0, 1], [1, 0 , 0, 57; 0, 1, 1, -1]], [[3, [3], [[3, 0; 0, 1]]], 2.71246530518434397468087 95106061300699, 1, [2, -1], [x + 7]], [Mat(1), [[0, 0]], [[1.790341756697729 3763292119206302198761, -1.7903417566977293763292119206302198761]]], [0, [Ma t([[6, 1]~, 1])]]], [[[25, 14; 0, 1], [1, 1]], [80, [20, 2, 2], [[2, 0]~, [- 24, 0]~, [4, 2]~]], Mat([[5, [-1, 1]~, 1, 1, [2, 1]~], 2]), [[[[4], [[2, 0]~ ], [[2, 0]~], [[0, 0]~], 1], [[5], [[6, 0]~], [[6, 0]~], [[0, 0]~], Mat([1/5 , -14/5])]], [[2, 2], [-24, [4, 2]~], [0, 1; 1, 1]]], [1, -12, 0, 0; 0, 0, 1 , 0; 0, 0, 0, 1]], [1], Mat([1, -3, -6, -6]), [12, [12], [[3, 0; 0, 1]]], [[ 0, 1, 0; -1/2, 5, 0], [-2, 0; 0, -10]]] ? bnrclassno(bnf,[[5,4;0,1],[1,0]]) 12 ? lu=ideallist(bnf,55,3); ? bnrclassnolist(bnf,lu) [[3], [], [3, 3], [3], [6, 6], [], [], [], [3, 3, 3], [], [3, 3], [3, 3], [] , [], [12, 6, 6, 12], [3], [3, 3], [], [9, 9], [6, 6], [], [], [], [], [6, 1 2, 6], [], [3, 3, 3, 3], [], [], [], [], [], [3, 6, 6, 3], [], [], [9, 3, 9] , [6, 6], [], [], [], [], [], [3, 3], [3, 3], [12, 12, 6, 6, 12, 12], [], [] , [6, 6], [9], [], [3, 3, 3, 3], [], [3, 3], [], [6, 12, 12, 6]] ? bnrdisc(bnr,Mat(6)) [12, 12, 18026977100265125] ? bnrdisc(bnr) [24, 12, 40621487921685401825918161408203125] ? bnrdisc(bnr2,,,2) 0 ? bnrdisc(bnr,Mat(6),,1) [6, 2, [125, 14; 0, 1]] ? bnrdisc(bnr,,,1) [12, 1, [1953125, 1160889; 0, 1]] ? bnrdisc(bnr2,,,3) 0 ? bnrdisclist(bnf,lu) [[[6, 6, Mat([229, 3])]], [], [[], []], [[]], [[12, 12, [5, 3; 229, 6]], [12 , 12, [5, 3; 229, 6]]], [], [], [], [[], [], []], [], [[], []], [[], []], [] , [], [[24, 24, [3, 6; 5, 9; 229, 12]], [], [], [24, 24, [3, 6; 5, 9; 229, 1 2]]], [[]], [[], []], [], [[18, 18, [19, 6; 229, 9]], [18, 18, [19, 6; 229, 9]]], [[], []], [], [], [], [], [[], [24, 24, [5, 12; 229, 12]], []], [], [[ ], [], [], []], [], [], [], [], [], [[], [12, 12, [3, 3; 11, 3; 229, 6]], [1 2, 12, [3, 3; 11, 3; 229, 6]], []], [], [], [[18, 18, [2, 12; 3, 12; 229, 9] ], [], [18, 18, [2, 12; 3, 12; 229, 9]]], [[12, 12, [37, 3; 229, 6]], [12, 1 2, [37, 3; 229, 6]]], [], [], [], [], [], [[], []], [[], []], [[], [], [], [ ], [], []], [], [], [[12, 12, [2, 12; 3, 3; 229, 6]], [12, 12, [2, 12; 3, 3; 229, 6]]], [[18, 18, [7, 12; 229, 9]]], [], [[], [], [], []], [], [[], []], [], [[], [24, 24, [5, 9; 11, 6; 229, 12]], [24, 24, [5, 9; 11, 6; 229, 12]] , []]] ? bnrdisclist(bnf,20) [[[[matrix(0,2), [[6, 6, Mat([229, 3])], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]], [], [[Mat([12, 1]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [12, 0, [3, 3; 229, 6 ]]]], [Mat([13, 1]), [[0, 0, 0], [12, 6, [-1, 1; 3, 3; 229, 6]], [0, 0, 0], [0, 0, 0]]]], [[Mat([10, 1]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]] , [[Mat([20, 1]), [[12, 12, [5, 3; 229, 6]], [0, 0, 0], [0, 0, 0], [24, 0, [ 5, 9; 229, 12]]]], [Mat([21, 1]), [[12, 12, [5, 3; 229, 6]], [24, 12, [5, 9; 229, 12]], [0, 0, 0], [0, 0, 0]]]], [], [], [], [[Mat([12, 2]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]], [[12, 1; 13, 1], [[0, 0, 0], [0, 0, 0], [12, 6, [-1, 1; 3, 6; 229, 6]], [24, 0, [3, 12; 229, 12]]]], [Mat([13, 2]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]], [], [[Mat([44, 1]), [[0, 0, 0], [12, 6, [-1, 1; 11, 3; 229, 6]], [0, 0, 0], [0, 0, 0]]], [Mat([45, 1]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [12, 0, [11, 3; 229, 6]]]]], [[[10, 1; 12, 1], [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]], [[10, 1; 13, 1], [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]], [], [], [[[12, 1; 20, 1], [[24, 24, [3, 6; 5, 9; 229, 12]], [0, 0, 0], [0, 0, 0], [48, 0, [3, 12; 5, 18; 229, 2 4]]]], [[13, 1; 20, 1], [[0, 0, 0], [24, 12, [3, 6; 5, 6; 229, 12]], [24, 12 , [3, 6; 5, 9; 229, 12]], [48, 0, [3, 12; 5, 18; 229, 24]]]], [[12, 1; 21, 1 ], [[0, 0, 0], [0, 0, 0], [24, 12, [3, 6; 5, 9; 229, 12]], [48, 0, [3, 12; 5 , 18; 229, 24]]]], [[13, 1; 21, 1], [[24, 24, [3, 6; 5, 9; 229, 12]], [48, 2 4, [3, 12; 5, 18; 229, 24]], [0, 0, 0], [0, 0, 0]]]], [[Mat([10, 2]), [[0, 0 , 0], [12, 6, [-1, 1; 2, 12; 229, 6]], [12, 6, [-1, 1; 2, 12; 229, 6]], [24, 0, [2, 36; 229, 12]]]]], [[Mat([68, 1]), [[0, 0, 0], [0, 0, 0], [12, 6, [-1 , 1; 17, 3; 229, 6]], [0, 0, 0]]], [Mat([69, 1]), [[0, 0, 0], [0, 0, 0], [12 , 6, [-1, 1; 17, 3; 229, 6]], [0, 0, 0]]]], [], [[Mat([76, 1]), [[18, 18, [1 9, 6; 229, 9]], [0, 0, 0], [0, 0, 0], [36, 0, [19, 15; 229, 18]]]], [Mat([77 , 1]), [[18, 18, [19, 6; 229, 9]], [36, 18, [-1, 1; 19, 15; 229, 18]], [0, 0 , 0], [0, 0, 0]]]], [[[10, 1; 20, 1], [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]], [[10, 1; 21, 1], [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]]]] ? bnrisprincipal(bnr,idealprimedec(bnf,7)[1]) [[9]~, [112595/19683, 13958/19683]~] ? dirzetak(nf4,30) [1, 2, 0, 3, 1, 0, 0, 4, 0, 2, 1, 0, 0, 0, 0, 5, 1, 0, 0, 3, 0, 2, 0, 0, 2, 0, 1, 0, 1, 0] ? factornf(x^3+x^2-2*x-1,t^3+t^2-2*t-1) [Mod(1, t^3 + t^2 - 2*t - 1)*x + Mod(-t, t^3 + t^2 - 2*t - 1) 1] [Mod(1, t^3 + t^2 - 2*t - 1)*x + Mod(-t^2 + 2, t^3 + t^2 - 2*t - 1) 1] [Mod(1, t^3 + t^2 - 2*t - 1)*x + Mod(t^2 + t - 1, t^3 + t^2 - 2*t - 1) 1] ? vp=idealprimedec(nf,3)[1] [3, [1, 0, 1, 0, 0]~, 1, 1, [1, -1, -1, -1, 0]~] ? idx=idealmul(nf,matid(5),vp) [3 2 1 0 1] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? idealinv(nf,idx) [1 0 0 2/3 0] [0 1 0 1/3 0] [0 0 1 1/3 0] [0 0 0 1/3 0] [0 0 0 0 1] ? idy=idealred(nf,idx,[1,5,6]) [5 0 0 0 2] [0 5 0 0 2] [0 0 5 0 1] [0 0 0 5 2] [0 0 0 0 1] ? idx2=idealmul(nf,idx,idx) [9 5 7 0 4] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? idt=idealmul(nf,idx,idx,1) [2 0 0 0 0] [0 2 0 0 0] [0 0 2 0 0] [0 0 0 2 1] [0 0 0 0 1] ? idz=idealintersect(nf,idx,idy) [15 10 5 0 12] [0 5 0 0 2] [0 0 5 0 1] [0 0 0 5 2] [0 0 0 0 1] ? aid=[idx,idy,idz,matid(5),idx] [[3, 2, 1, 0, 1; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1] , [5, 0, 0, 0, 2; 0, 5, 0, 0, 2; 0, 0, 5, 0, 1; 0, 0, 0, 5, 2; 0, 0, 0, 0, 1 ], [15, 10, 5, 0, 12; 0, 5, 0, 0, 2; 0, 0, 5, 0, 1; 0, 0, 0, 5, 2; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [3, 2, 1, 0, 1; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0 , 0, 1]] ? bid=idealstar(nf2,54,1) [[[54, 0, 0; 0, 54, 0; 0, 0, 54], [0]], [132678, [1638, 9, 9]], [[2, [2, 0, 0]~, 1, 3, [1, 0, 0]~], 1; [3, [3, 0, 0]~, 1, 3, [1, 0, 0]~], 3], [[[[7], [[ 1, 1, 1]~], [[1, -27, -27]~], [[]~], 1]], [[[26], [[4, 2, 2]~], [[-23, 2, 2] ~], [[]~], 1], [[3, 3, 3], [[4, 0, 0]~, [1, 3, 0]~, [1, 0, 3]~], [[-23, 0, 0 ]~, [1, -24, 0]~, [1, 0, -24]~], [[]~, []~, []~], [1/3, 0, 0; 0, 1/3, 0; 0, 0, 1/3]], [[3, 3, 3], [[10, 0, 0]~, [1, 9, 0]~, [1, 0, 9]~], [[-17, 0, 0]~, [1, -18, 0]~, [1, 0, -18]~], [[]~, []~, []~], [1/9, 0, 0; 0, 1/9, 0; 0, 0, 1 /9]]], [[], [], [;]]], [468, 469, 0, 0, -364, 0, 0, -1092; 0, 0, 1, 0, -6, - 6, 0, 0; 0, 0, 0, 1, -3, 0, -6, 0]] ? vaid=[idx,idy,matid(5)] [[3, 2, 1, 0, 1; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1] , [5, 0, 0, 0, 2; 0, 5, 0, 0, 2; 0, 0, 5, 0, 1; 0, 0, 0, 5, 2; 0, 0, 0, 0, 1 ], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]] ? haid=[matid(5),matid(5),matid(5)] [[1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1 ], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]] ? idealadd(nf,idx,idy) [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? idealaddtoone(nf,idx,idy) [[3, 2, 1, -3, 1]~, [-2, -2, -1, 3, -1]~] ? idealaddtoone(nf,[idy,idx]) [[-5, 0, 0, 0, 0]~, [6, 0, 0, 0, 0]~] ? idealappr(nf,idy) [-1, -1, 2, 4, -3]~ ? idealappr(nf,idealfactor(nf,idy),1) [-1, -1, 2, 4, -3]~ ? idealcoprime(nf,idx,idx) [-1/3, 1/3, 1/3, 1/3, 0]~ ? idealdiv(nf,idy,idt) [5 0 5/2 0 1] [0 5/2 0 0 1] [0 0 5/2 0 1/2] [0 0 0 5/2 1] [0 0 0 0 1/2] ? idealdiv(nf,idx2,idx,1) [3 2 1 0 1] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? idf=idealfactor(nf,idz) [[3, [1, 0, 1, 0, 0]~, 1, 1, [1, -1, -1, -1, 0]~] 1] [[5, [-1, 0, 0, 0, 2]~, 4, 1, [2, 2, 1, 2, 1]~] 3] [[5, [2, 0, 0, 0, -2]~, 1, 1, [2, 0, 3, 0, 1]~] 1] ? idealhnf(nf,vp) [3 2 1 0 1] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? idealhnf(nf,vp[2],3) [3 2 1 0 1] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? ideallist(bnf,20) [[[1, 0; 0, 1]], [], [[3, 0; 0, 1], [3, 1; 0, 1]], [[2, 0; 0, 2]], [[5, 4; 0 , 1], [5, 2; 0, 1]], [], [], [], [[9, 6; 0, 1], [3, 0; 0, 3], [9, 4; 0, 1]], [], [[11, 10; 0, 1], [11, 2; 0, 1]], [[6, 0; 0, 2], [6, 2; 0, 2]], [], [], [[15, 9; 0, 1], [15, 4; 0, 1], [15, 12; 0, 1], [15, 7; 0, 1]], [[4, 0; 0, 4] ], [[17, 15; 0, 1], [17, 3; 0, 1]], [], [[19, 0; 0, 1], [19, 1; 0, 1]], [[10 , 8; 0, 2], [10, 4; 0, 2]]] ? ideallog(nf2,w,bid) [866, 8, 0]~ ? idealmin(nf,idx,[1,2,3]) [[-2; 1; 1; 0; 1], [2.0885812311199768913287869744681966009 + 3.141592653589 7932384626433832795028842*I, 1.5921096812520196555597562531657929785 + 4.244 7196639216499665715751642189271111*I, -0.79031915447583185468082063233076160 208 + 2.5437460822678889883600220330800078854*I]] ? idealnorm(nf,idt) 16 ? idp=idealpow(nf,idx,7) [2187 1436 1807 630 1822] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? idealpow(nf,idx,7,1) [1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1] ? idealprimedec(nf,2) [[2, [3, 0, 1, 0, 0]~, 1, 1, [0, 0, 0, 1, 1]~], [2, [12, -4, -2, 11, 3]~, 1, 4, [1, 0, 1, 0, 0]~]] ? idealprimedec(nf,3) [[3, [1, 0, 1, 0, 0]~, 1, 1, [1, -1, -1, -1, 0]~], [3, [1, 1, 1, 0, 0]~, 2, 2, [0, 2, 2, 1, 0]~]] ? idealprimedec(nf,11) [[11, [11, 0, 0, 0, 0]~, 1, 5, [1, 0, 0, 0, 0]~]] ? idealprincipal(nf,Mod(x^3+5,nfpol)) [6] [1] [3] [1] [3] ? idealtwoelt(nf,idy) [5, [2, 2, 1, 2, 1]~] ? idealtwoelt(nf,idy,10) [-1, -1, 2, -1, 2]~ ? idealstar(nf2,54) [[[54, 0, 0; 0, 54, 0; 0, 0, 54], [0]], [132678, [1638, 9, 9]], [[2, [2, 0, 0]~, 1, 3, [1, 0, 0]~], 1; [3, [3, 0, 0]~, 1, 3, [1, 0, 0]~], 3], [[[[7], [[ 0, 0, 1]~], [[-26, 0, -27]~], [[]~], 1]], [[[26], [[4, 2, 1]~], [[-23, 2, -2 6]~], [[]~], 1], [[3, 3, 3], [[4, 0, 0]~, [1, 3, 0]~, [1, 0, 3]~], [[-23, 0, 0]~, [1, -24, 0]~, [1, 0, -24]~], [[]~, []~, []~], [1/3, 0, 0; 0, 1/3, 0; 0 , 0, 1/3]], [[3, 3, 3], [[10, 0, 0]~, [1, 9, 0]~, [1, 0, 9]~], [[-17, 0, 0]~ , [1, -18, 0]~, [1, 0, -18]~], [[]~, []~, []~], [1/9, 0, 0; 0, 1/9, 0; 0, 0, 1/9]]], [[], [], [;]]], [468, -77, 0, 728, -1456, 0, 546, -1092; 0, 0, 1, 0 , -1, -6, 0, -3; 0, 1, 0, -1, 1, 0, -3, 3]] ? idealval(nf,idp,vp) 7 ? ideleprincipal(nf,Mod(x^3+5,nfpol)) [[6; 1; 3; 1; 3], [2.2324480827796254080981385584384939685 + 3.1415926535897 932384626433832795028842*I, 5.0387659675158716386435353106610489968 + 1.5851 760343512250049897278861965702424*I, 4.2664040272651028743625910797589683173 - 0.0083630478144368246110910258645462996337*I]] ? ba=nfalgtobasis(nf,Mod(x^3+5,nfpol)) [6, 1, 3, 1, 3]~ ? bb=nfalgtobasis(nf,Mod(x^3+x,nfpol)) [1, 1, 4, 1, 3]~ ? bc=matalgtobasis(nf,[Mod(x^2+x,nfpol);Mod(x^2+1,nfpol)]) [[3, -2, 1, 1, 0]~] [[4, -2, 0, 1, 0]~] ? matbasistoalg(nf,bc) [Mod(x^2 + x, x^5 - 5*x^3 + 5*x + 25)] [Mod(x^2 + 1, x^5 - 5*x^3 + 5*x + 25)] ? nfbasis(x^3+4*x+5) [1, x, 1/7*x^2 - 1/7*x - 2/7] ? nfbasis(x^3+4*x+5,2) [1, x, 1/7*x^2 - 1/7*x - 2/7] ? nfbasis(x^3+4*x+12,1) [1, x, 1/2*x^2] ? nfbasistoalg(nf,ba) Mod(x^3 + 5, x^5 - 5*x^3 + 5*x + 25) ? nfbasis(p2,0,fa) [1, x, x^2, 1/11699*x^3 + 1847/11699*x^2 - 132/11699*x - 2641/11699, 1/13962 3738889203638909659*x^4 - 1552451622081122020/139623738889203638909659*x^3 + 418509858130821123141/139623738889203638909659*x^2 - 6810913798507599407313 4/139623738889203638909659*x - 13185339461968406/58346808996920447] ? da=nfdetint(nf,[a,aid]) [90 70 35 0 65] [0 5 0 0 0] [0 0 5 0 0] [0 0 0 5 0] [0 0 0 0 5] ? nfdisc(x^3+4*x+12) -1036 ? nfdisc(x^3+4*x+12,1) -1036 ? nfdisc(p2,0,fa) 136866601 ? nfeltdiv(nf,ba,bb) [584/373, 66/373, -32/373, -105/373, 120/373]~ ? nfeltdiveuc(nf,ba,bb) [2, 0, 0, 0, 0]~ ? nfeltdivrem(nf,ba,bb) [[2, 0, 0, 0, 0]~, [4, -1, -5, -1, -3]~] ? nfeltmod(nf,ba,bb) [4, -1, -5, -1, -3]~ ? nfeltmul(nf,ba,bb) [50, -15, -35, 60, 15]~ ? nfeltpow(nf,bb,5) [-291920, 136855, 230560, -178520, 74190]~ ? nfeltreduce(nf,ba,idx) [1, 0, 0, 0, 0]~ ? nfeltval(nf,ba,vp) 0 ? nffactor(nf2,x^3+x) [x 1] [x^2 + 1 1] ? aut=nfgaloisconj(nf3) [-x, x, -1/12*x^4 - 1/2*x, -1/12*x^4 + 1/2*x, 1/12*x^4 - 1/2*x, 1/12*x^4 + 1 /2*x]~ ? nfgaloisapply(nf3,aut[5],Mod(x^5,x^6+108)) Mod(-1/2*x^5 + 9*x^2, x^6 + 108) ? nfhilbert(nf,3,5) -1 ? nfhilbert(nf,3,5,idf[1,1]) -1 ? nfhnf(nf,[a,aid]) [[[1, 0, 0, 0, 0]~, [-2, 0, 0, 0, 0]~, [0, 0, 0, 0, 0]~; [0, 0, 0, 0, 0]~, [ 1, 0, 0, 0, 0]~, [0, 0, 0, 0, 0]~; [0, 0, 0, 0, 0]~, [0, 0, 0, 0, 0]~, [1, 0 , 0, 0, 0]~], [[10, 4, 5, 6, 7; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0 ; 0, 0, 0, 0, 1], [3, 2, 1, 0, 1; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]]] ? nfhnfmod(nf,[a,aid],da) [[[1, 0, 0, 0, 0]~, [-2, 0, 0, 0, 0]~, [0, 0, 0, 0, 0]~; [0, 0, 0, 0, 0]~, [ 1, 0, 0, 0, 0]~, [0, 0, 0, 0, 0]~; [0, 0, 0, 0, 0]~, [0, 0, 0, 0, 0]~, [1, 0 , 0, 0, 0]~], [[10, 4, 5, 6, 7; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0 ; 0, 0, 0, 0, 1], [3, 2, 1, 0, 1; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]]] ? nfisideal(bnf[7],[5,2;0,1]) 1 ? nfisincl(x^2+1,x^4+1) [-x^2, x^2] ? nfisincl(x^2+1,nfinit(x^4+1)) [-x^2, x^2] ? nfisisom(x^3+x^2-2*x-1,x^3+x^2-2*x-1) [x, -x^2 - x + 1, x^2 - 2] ? nfisisom(x^3-2,nfinit(x^3-6*x^2-6*x-30)) [-1/25*x^2 + 13/25*x - 2/5] ? nfroots(nf2,x+2) [Mod(-2, y^3 - y - 1)] ? nfrootsof1(nf) [2, [-1, 0, 0, 0, 0]~] ? nfsnf(nf,[as,haid,vaid]) [[2562748315629757085585610, 436545976069778274371140, 123799938628701108220 1405, 2356446991473627724963350, 801407102592194537169612; 0, 5, 0, 0, 2; 0, 0, 5, 0, 1; 0, 0, 0, 5, 2; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0 , 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]] ? nfsubfields(nf) [[x^5 - 5*x^3 + 5*x + 25, x], [x, x^5 - 5*x^3 + 5*x + 25]] ? polcompositum(x^4-4*x+2,x^3-x-1) [x^12 - 4*x^10 + 8*x^9 + 12*x^8 + 12*x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58*x ^2 - 128*x - 5] ? polcompositum(x^4-4*x+2,x^3-x-1,1) [[x^12 - 4*x^10 + 8*x^9 + 12*x^8 + 12*x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58* x^2 - 128*x - 5, Mod(-279140305176/29063006931199*x^11 + 129916611552/290630 06931199*x^10 + 1272919322296/29063006931199*x^9 - 2813750209005/29063006931 199*x^8 - 2859411937992/29063006931199*x^7 - 414533880536/29063006931199*x^6 - 35713977492936/29063006931199*x^5 - 17432607267590/29063006931199*x^4 + 4 9785595543672/29063006931199*x^3 + 9423768373204/29063006931199*x^2 - 427797 76146743/29063006931199*x + 37962587857138/29063006931199, x^12 - 4*x^10 + 8 *x^9 + 12*x^8 + 12*x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58*x^2 - 128*x - 5), M od(-279140305176/29063006931199*x^11 + 129916611552/29063006931199*x^10 + 12 72919322296/29063006931199*x^9 - 2813750209005/29063006931199*x^8 - 28594119 37992/29063006931199*x^7 - 414533880536/29063006931199*x^6 - 35713977492936/ 29063006931199*x^5 - 17432607267590/29063006931199*x^4 + 49785595543672/2906 3006931199*x^3 + 9423768373204/29063006931199*x^2 - 13716769215544/290630069 31199*x + 37962587857138/29063006931199, x^12 - 4*x^10 + 8*x^9 + 12*x^8 + 12 *x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58*x^2 - 128*x - 5), -1]] ? polgalois(x^6-3*x^2-1) [12, 1, 1, "A_4(6) = [2^2]3"] ? polred(x^5-2*x^4-4*x^3-96*x^2-352*x-568) [x - 1, x^5 - x^4 - 6*x^3 + 6*x^2 + 13*x - 5, x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1, x^5 - x^4 + 4*x^3 - 2*x^2 + x - 1, x^5 + 4*x^3 - 4*x^2 + 8*x - 8] ? polred(x^4-28*x^3-458*x^2+9156*x-25321,3) [1 x - 1] [1/115*x^2 - 14/115*x - 327/115 x^2 - 10] [3/1495*x^3 - 63/1495*x^2 - 1607/1495*x + 13307/1495 x^4 - 32*x^2 + 216] [1/4485*x^3 - 7/1495*x^2 - 1034/4485*x + 7924/4485 x^4 - 8*x^2 + 6] ? polred(x^4+576,1) [x - 1, x^2 - x + 1, x^2 + 1, x^4 - x^2 + 1] ? polred(x^4+576,3) [1 x - 1] [-1/192*x^3 - 1/8*x + 1/2 x^2 - x + 1] [1/24*x^2 x^2 + 1] [1/192*x^3 - 1/48*x^2 - 1/8*x x^4 - x^2 + 1] ? polred(p2,0,fa) [x - 1, x^5 - 2*x^4 - 62*x^3 + 85*x^2 + 818*x + 1, x^5 - 2*x^4 - 53*x^3 - 46 *x^2 + 508*x + 913, x^5 - 2*x^4 - 13*x^3 + 37*x^2 - 21*x - 1, x^5 - x^4 - 52 *x^3 - 197*x^2 - 273*x - 127] ? polred(p2,1,fa) [x - 1, x^5 - 2*x^4 - 62*x^3 + 85*x^2 + 818*x + 1, x^5 - 2*x^4 - 53*x^3 - 46 *x^2 + 508*x + 913, x^5 - 2*x^4 - 13*x^3 + 37*x^2 - 21*x - 1, x^5 - x^4 - 52 *x^3 - 197*x^2 - 273*x - 127] ? polredabs(x^5-2*x^4-4*x^3-96*x^2-352*x-568) x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1 ? polredabs(x^5-2*x^4-4*x^3-96*x^2-352*x-568,1) [x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1, Mod(2*x^4 - x^3 + 3*x^2 - 3*x - 1, x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1)] ? polredord(x^3-12*x+45*x-1) [x - 1, x^3 - 363*x - 2663, x^3 + 33*x - 1] ? polsubcyclo(31,5) x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5 ? setrand(1);poltschirnhaus(x^5-x-1) x^5 - 15*x^4 + 88*x^3 - 278*x^2 + 452*x - 289 ? aa=rnfpseudobasis(nf2,p) [[[1, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [-2, 0, 0]~, [3, 1, 0]~; [0, 0, 0]~, [ 1, 0, 0]~, [0, 0, 0]~, [2, 0, 0]~, [0, -1, 0]~; [0, 0, 0]~, [0, 0, 0]~, [1, 0, 0]~, [1, 0, 0]~, [-5, -2, 0]~; [0, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [1, 0, 0]~, [-2, 0, 0]~; [0, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [1, 0, 0] ~], [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 2/5; 0, 1, 3/5; 0, 0, 1/5], [1, 0, 22/25; 0, 1, 8/25; 0, 0, 1/25]], [416134375, 202396875, 60056800; 0, 3125, 2700; 0, 0, 25], [- 1275, 5, 5]~] ? rnfbasis(bnf2,aa) [[1, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [-26/25, 11/25, -8/25]~ [0, 4, -7]~] [[0, 0, 0]~ [1, 0, 0]~ [0, 0, 0]~ [53/25, -8/25, -1/25]~ [6/5, -41/5, 53/5]~ ] [[0, 0, 0]~ [0, 0, 0]~ [1, 0, 0]~ [-14/25, -21/25, 13/25]~ [-16/5, 1/5, 7/5] ~] [[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [7/25, -2/25, 6/25]~ [2/5, -2/5, 11/5]~] [[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [9/25, 1/25, -3/25]~ [2/5, -7/5, 6/5]~] ? rnfdisc(nf2,p) [[416134375, 202396875, 60056800; 0, 3125, 2700; 0, 0, 25], [-1275, 5, 5]~] ? rnfequation(nf2,p) x^15 - 15*x^11 + 75*x^7 - x^5 - 125*x^3 + 5*x + 1 ? rnfequation(nf2,p,1) [x^15 - 15*x^11 + 75*x^7 - x^5 - 125*x^3 + 5*x + 1, Mod(-x^5 + 5*x, x^15 - 1 5*x^11 + 75*x^7 - x^5 - 125*x^3 + 5*x + 1), 0] ? rnfhnfbasis(bnf2,aa) [[1, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [-6/5, -4/5, 2/5]~ [3/25, -8/25, 24/25]~] [[0, 0, 0]~ [1, 0, 0]~ [0, 0, 0]~ [6/5, 4/5, -2/5]~ [-9/25, -1/25, 3/25]~] [[0, 0, 0]~ [0, 0, 0]~ [1, 0, 0]~ [3/5, 2/5, -1/5]~ [-8/25, 13/25, -39/25]~] [[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [3/5, 2/5, -1/5]~ [4/25, 6/25, -18/25]~] [[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [-2/25, -3/25, 9/25]~] ? rnfisfree(bnf2,aa) 1 ? rnfsteinitz(nf2,aa) [[[1, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [-26/25, 11/25, -8/25]~, [29/125, -2/2 5, 8/125]~; [0, 0, 0]~, [1, 0, 0]~, [0, 0, 0]~, [53/25, -8/25, -1/25]~, [-53 /125, 7/125, 1/125]~; [0, 0, 0]~, [0, 0, 0]~, [1, 0, 0]~, [-14/25, -21/25, 1 3/25]~, [9/125, 19/125, -13/125]~; [0, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [7/25 , -2/25, 6/25]~, [-9/125, 2/125, -6/125]~; [0, 0, 0]~, [0, 0, 0]~, [0, 0, 0] ~, [9/25, 1/25, -3/25]~, [-8/125, -1/125, 3/125]~], [[1, 0, 0; 0, 1, 0; 0, 0 , 1], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0; 0, 1, 0; 0, 0, 1], [125, 0, 22; 0, 125, 108; 0, 0, 1]], [416134375, 202396875, 60056800; 0, 3125, 2700; 0, 0, 25], [-1275, 5, 5]~] ? nfz=zetakinit(x^2-2); ? zetak(nfz,-3) 0.091666666666666666666666666666666666667 ? zetak(nfz,1.5+3*I) 0.88324345992059326405525724366416928893 - 0.2067536250233895222724230899142 7938843*I ? setrand(1);quadclassunit(1-10^7,,[1,1]) [2416, [1208, 2], [Qfb(277, 55, 9028), Qfb(1700, 1249, 1700)], 1] ? setrand(1);quadclassunit(10^9-3,,[0.5,0.5]) [4, [4], [Qfb(211, 31405, -16263, 0.E-48)], 2800.625251907016076486370621737 0745514] ? sizebyte(%) 152 ? getheap [200, 121341] ? print("Total time spent: ",gettime); Total time spent: 288 ? \q