? ?    realprecision = 38 significant digits
?    echo = 1 (on)
? ? 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+385389051407205
7
% = x^5 + 3021*x^4 - 786303*x^3 - 6826636057*x^2 - 546603588746*x + 385389
0514072057
? 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, -2.42851749071941860
68992069565359418364, 5.8976972027301414394898806541072047941, -7.07345267
15090929269887668671457811020, 3.8085820570096366144649278594400435257; 1,
 1.9647119211288133163138753392090569931 + 0.80971492418897895128294082219
556466857*I, 3.2044546745713084269203768790545260356 + 3.18171312854000053
41145852263331539899*I, -0.16163499313031744537610982231988834519 + 1.8880
437862007056931906454476483475283*I, 2.06607095383724806326989711488010906
92 + 2.6898967519623140991170523711857387388*I; 1, -0.75045317576910401286
427186094108607489 + 1.3101462685358123283560773619310445915*I, -1.1533032
759363791466653172061081284327 - 1.9664068558894834311780119356739268309*I
, 1.1983613288848639088704932558927788962 + 0.6437023807625698889957032567
1192132449*I, -0.47036198234206637050236104460013083212 + 0.08362826671158
9186119416762685933385421*I], [1, 2, 2; -2.4285174907194186068992069565359
418364, 3.9294238422576266326277506784181139862 - 1.6194298483779579025658
816443911293371*I, -1.5009063515382080257285437218821721497 - 2.6202925370
716246567121547238620891831*I; 5.8976972027301414394898806541072047941, 6.
4089093491426168538407537581090520712 - 6.36342625708000106822917045266630
79798*I, -2.3066065518727582933306344122162568654 + 3.93281371177896686235
60238713478536619*I; -7.0734526715090929269887668671457811020, -0.32326998
626063489075221964463977669038 - 3.7760875724014113863812908952966950567*I
, 2.3967226577697278177409865117855577924 - 1.2874047615251397779914065134
238426489*I; 3.8085820570096366144649278594400435257, 4.132141907674496126
5397942297602181385 - 5.3797935039246281982341047423714774776*I, -0.940723
96468413274100472208920026166424 - 0.1672565334231783722388335253718667708
4*I], [5, 4.0215293653309345240000000000000000000 E-87, 10.000000000000000
000000000000000000000, -5.0000000000000000000000000000000000000, 7.0000000
000000000000000000000000000000; 4.0215293653309345240000000000000000000 E-
87, 19.488486013650707197449403270536023970, 8.043058730661869049000000000
0000000000 E-86, 19.488486013650707197449403270536023970, 4.15045922467060
85588902013976045703227; 10.000000000000000000000000000000000000, 8.043058
7306618690490000000000000000000 E-86, 85.960217420851846480305133936577594
605, -36.034268291482979838267056239752434596, 53.576130452511107888183080
361946556763; -5.0000000000000000000000000000000000000, 19.488486013650707
197449403270536023970, -36.034268291482979838267056239752434596, 60.916248
374441986300937507618575151517, -18.470101750219179344070032346246890434; 
7.0000000000000000000000000000000000000, 4.1504592246706085588902013976045
703227, 53.576130452511107888183080361946556763, -18.470101750219179344070
032346246890434, 37.970152892842367340897384258599214282], [5, 0, 10, -5, 
7; 0, 10, 0, 10, -5; 10, 0, 30, -55, 20; -5, 10, -55, 45, -39; 7, -5, 20, 
-39, 9], [345, 0, 340, 167, 150; 0, 345, 110, 220, 153; 0, 0, 5, 2, 1; 0, 
0, 0, 1, 0; 0, 0, 0, 0, 1], [132825, -18975, -5175, 27600, 17250; -18975, 
34500, 41400, 3450, -43125; -5175, 41400, -41400, -15525, 51750; 27600, 34
50, -15525, -3450, 0; 17250, -43125, 51750, 0, -86250], [595125, 0, 0, 581
325, 474375; 0, 119025, 0, 117300, 63825; 0, 0, 119025, 67275, 113850; 0, 
0, 0, 1725, 0; 0, 0, 0, 0, 8625]], [-2.42851749071941860689920695653594183
64, 1.9647119211288133163138753392090569931 + 0.80971492418897895128294082
219556466857*I, -0.75045317576910401286427186094108607489 + 1.310146268535
8123283560773619310445915*I], [1, x, x^2, 1/3*x^3 - 1/3*x^2 - 1/3, 1/15*x^
4 + 1/3*x^2 + 1/3*x + 1/3], [1, 0, 0, 1, -5; 0, 1, 0, 0, -5; 0, 0, 1, 1, -
5; 0, 0, 0, 3, 0; 0, 0, 0, 0, 15], [1, 0, 0, 0, 0, 0, 0, 1, -2, -1, 0, 1, 
-5, -5, -3, 0, -2, -5, 1, -4, 0, -1, -3, -4, -3; 0, 1, 0, 0, 0, 1, 0, 0, -
2, 0, 0, 0, -5, 0, -5, 0, -2, 0, -5, 0, 0, 0, -5, 0, -4; 0, 0, 1, 0, 0, 0,
 1, 1, -2, 1, 1, 1, -5, 3, -3, 0, -2, 3, -5, 1, 0, 1, -3, 1, -2; 0, 0, 0, 
1, 0, 0, 0, 3, -1, 2, 0, 3, 0, 5, 1, 1, -1, 5, -4, 3, 0, 2, 1, 3, 1; 0, 0,
 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 15, -5, 10, 0, 5, -5, 10, -2, 1, 0, 10, -2,
 7]]
? nf1=nfinit(nfpol,2)
% = [x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2, [1, 2], 595125, 4, [[1, -1.089
1151457205048250249527946671612684, 1.186171800637796459479629386048398986
0, -0.59741050929194782733001765987770358483, 0.15894419745390376206549481
671071894289; 1, -0.13838372073406036365047976417441696637 + 0.49181637657
768643499753285514741525107*I, -0.22273329410580226599155701611419649154 -
 0.13611876021752805221674918029071012580*I, -0.13167445871785818798769651
537619416009 + 0.13249517760521973840801462296650806543*I, -0.053650958656
997725359297528357602608116 + 0.27622636814169107038138284681568361486*I; 
1, 1.6829412935943127761629561615079976005 + 2.050035122601072617297428698
3598602163*I, -1.3703526062130959637482576769100030014 + 6.900177522288049
4773720769629846373016*I, -8.0696202866361678983472946546849540475 + 8.876
7676785971042450885284301348051602*I, -22.02582114006995415567344987999775
6863 - 8.4306586896999153544710860185447589664*I], [1, 2, 2; -1.0891151457
205048250249527946671612684, -0.27676744146812072730095952834883393274 - 0
.98363275315537286999506571029483050214*I, 3.36588258718862555232591232301
59952011 - 4.1000702452021452345948573967197204327*I; 1.186171800637796459
4796293860483989860, -0.44546658821160453198311403222839298308 + 0.2722375
2043505610443349836058142025160*I, -2.740705212426191927496515353820006002
9 - 13.800355044576098954744153925969274603*I; -0.597410509291947827330017
65987770358483, -0.26334891743571637597539303075238832018 - 0.264990355210
43947681602924593301613087*I, -16.139240573272335796694589309369908095 - 1
7.753535357194208490177056860269610320*I; 0.158944197453903762065494816710
71894289, -0.10730191731399545071859505671520521623 - 0.552452736283382140
76276569363136722973*I, -44.051642280139908311346899759995513726 + 16.8613
17379399830708942172037089517932*I], [5, 2.0000000000000000000000000000000
000000, -2.0000000000000000000000000000000000000, -17.00000000000000000000
0000000000000000, -44.000000000000000000000000000000000000; 2.000000000000
0000000000000000000000000, 15.778109408671998044836357471283695361, 22.314
643349754061651916553814602769764, 10.051395257831478275499932716306366248
, -108.58917507620841447456569092094763671; -2.000000000000000000000000000
0000000000, 22.314643349754061651916553814602769764, 100.52391262388960975
827806174040462368, 143.93295090847353519436673793501057176, -55.842564718
082452641322500190813370023; -17.000000000000000000000000000000000000, 10.
051395257831478275499932716306366248, 143.93295090847353519436673793501057
176, 288.25823756749944693139292174819167135, 205.798400382776623757201806
49465932302; -44.000000000000000000000000000000000000, -108.58917507620841
447456569092094763671, -55.842564718082452641322500190813370023, 205.79840
038277662375720180649465932302, 1112.6092277946777707779250962522343036], 
[5, 2, -2, -17, -44; 2, -2, -34, -63, -40; -2, -34, -90, -101, 177; -17, -
63, -101, -27, 505; -44, -40, 177, 505, 828], [345, 0, 160, 252, 156; 0, 3
45, 215, 311, 306; 0, 0, 5, 3, 2; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [163875, 
-388125, -296700, 234600, -89700; -388125, -1593900, -677925, 595125, -315
675; -296700, -677925, 17250, 58650, -87975; 234600, 595125, 58650, -10005
0, 89700; -89700, -315675, -87975, 89700, -55200], [595125, 0, 357075, 427
800, 326025; 0, 595125, 119025, 512325, 191475; 0, 0, 119025, 79350, 75900
; 0, 0, 0, 1725, 0; 0, 0, 0, 0, 1725]], [-1.089115145720504825024952794667
1612684, -0.13838372073406036365047976417441696637 + 0.4918163765776864349
9753285514741525107*I, 1.6829412935943127761629561615079976005 + 2.0500351
226010726172974286983598602163*I], [1, x, x^2, 1/2*x^3 + 1/2*x^2 + 1/2*x, 
1/2*x^4 + 1/2*x], [1, 0, 0, 0, 0; 0, 1, 0, -1, -1; 0, 0, 1, -1, 0; 0, 0, 0
, 2, 0; 0, 0, 0, 0, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -2, 0
, 0, -1, -2, -2, 0, -1, -2, -2, 5; 0, 1, 0, 0, 0, 1, 0, -1, -1, -1, 0, -1,
 -1, -2, 2, 0, -1, -2, -1, 7, 0, -1, 2, 7, 14; 0, 0, 1, 0, 0, 0, 1, -1, 0,
 -2, 1, -1, 0, -3, -3, 0, 0, -3, -4, -1, 0, -2, -3, -1, 15; 0, 0, 0, 1, 0,
 0, 0, 2, 1, -3, 0, 2, 0, -2, -13, 1, 1, -2, -9, -19, 0, -3, -13, -19, 7; 
0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 1, 3, 4, -4, 1, 2, 1, -4, 
-21]]
? nfinit(nfpol,3)
% = [[x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2, [1, 2], 595125, 4, [[1, -1.08
91151457205048250249527946671612684, 1.18617180063779645947962938604839898
60, -0.59741050929194782733001765987770358483, 0.1589441974539037620654948
1671071894289; 1, -0.13838372073406036365047976417441696637 + 0.4918163765
7768643499753285514741525107*I, -0.22273329410580226599155701611419649154 
- 0.13611876021752805221674918029071012580*I, -0.1316744587178581879876965
1537619416009 + 0.13249517760521973840801462296650806543*I, -0.05365095865
6997725359297528357602608116 + 0.27622636814169107038138284681568361486*I;
 1, 1.6829412935943127761629561615079976005 + 2.05003512260107261729742869
83598602163*I, -1.3703526062130959637482576769100030014 + 6.90017752228804
94773720769629846373016*I, -8.0696202866361678983472946546849540475 + 8.87
67676785971042450885284301348051602*I, -22.0258211400699541556734498799977
56863 - 8.4306586896999153544710860185447589664*I], [1, 2, 2; -1.089115145
7205048250249527946671612684, -0.27676744146812072730095952834883393274 - 
0.98363275315537286999506571029483050214*I, 3.3658825871886255523259123230
159952011 - 4.1000702452021452345948573967197204327*I; 1.18617180063779645
94796293860483989860, -0.44546658821160453198311403222839298308 + 0.272237
52043505610443349836058142025160*I, -2.74070521242619192749651535382000600
29 - 13.800355044576098954744153925969274603*I; -0.59741050929194782733001
765987770358483, -0.26334891743571637597539303075238832018 - 0.26499035521
043947681602924593301613087*I, -16.139240573272335796694589309369908095 - 
17.753535357194208490177056860269610320*I; 0.15894419745390376206549481671
071894289, -0.10730191731399545071859505671520521623 - 0.55245273628338214
076276569363136722973*I, -44.051642280139908311346899759995513726 + 16.861
317379399830708942172037089517932*I], [5, 2.000000000000000000000000000000
0000000, -2.0000000000000000000000000000000000000, -17.0000000000000000000
00000000000000000, -44.000000000000000000000000000000000000; 2.00000000000
00000000000000000000000000, 15.778109408671998044836357471283695361, 22.31
4643349754061651916553814602769764, 10.05139525783147827549993271630636624
8, -108.58917507620841447456569092094763671; -2.00000000000000000000000000
00000000000, 22.314643349754061651916553814602769764, 100.5239126238896097
5827806174040462368, 143.93295090847353519436673793501057176, -55.84256471
8082452641322500190813370023; -17.000000000000000000000000000000000000, 10
.051395257831478275499932716306366248, 143.9329509084735351943667379350105
7176, 288.25823756749944693139292174819167135, 205.79840038277662375720180
649465932302; -44.000000000000000000000000000000000000, -108.5891750762084
1447456569092094763671, -55.842564718082452641322500190813370023, 205.7984
0038277662375720180649465932302, 1112.6092277946777707779250962522343036],
 [5, 2, -2, -17, -44; 2, -2, -34, -63, -40; -2, -34, -90, -101, 177; -17, 
-63, -101, -27, 505; -44, -40, 177, 505, 828], [345, 0, 160, 252, 156; 0, 
345, 215, 311, 306; 0, 0, 5, 3, 2; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [163875,
 -388125, -296700, 234600, -89700; -388125, -1593900, -677925, 595125, -31
5675; -296700, -677925, 17250, 58650, -87975; 234600, 595125, 58650, -1000
50, 89700; -89700, -315675, -87975, 89700, -55200], [595125, 0, 357075, 42
7800, 326025; 0, 595125, 119025, 512325, 191475; 0, 0, 119025, 79350, 7590
0; 0, 0, 0, 1725, 0; 0, 0, 0, 0, 1725]], [-1.08911514572050482502495279466
71612684, -0.13838372073406036365047976417441696637 + 0.491816376577686434
99753285514741525107*I, 1.6829412935943127761629561615079976005 + 2.050035
1226010726172974286983598602163*I], [1, x, x^2, 1/2*x^3 + 1/2*x^2 + 1/2*x,
 1/2*x^4 + 1/2*x], [1, 0, 0, 0, 0; 0, 1, 0, -1, -1; 0, 0, 1, -1, 0; 0, 0, 
0, 2, 0; 0, 0, 0, 0, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -2, 
0, 0, -1, -2, -2, 0, -1, -2, -2, 5; 0, 1, 0, 0, 0, 1, 0, -1, -1, -1, 0, -1
, -1, -2, 2, 0, -1, -2, -1, 7, 0, -1, 2, 7, 14; 0, 0, 1, 0, 0, 0, 1, -1, 0
, -2, 1, -1, 0, -3, -3, 0, 0, -3, -4, -1, 0, -2, -3, -1, 15; 0, 0, 0, 1, 0
, 0, 0, 2, 1, -3, 0, 2, 0, -2, -13, 1, 1, -2, -9, -19, 0, -3, -13, -19, 7;
 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 1, 3, 4, -4, 1, 2, 1, -4,
 -21]], 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, -1.089
1151457205048250249527946671612684, 1.186171800637796459479629386048398986
0, -0.59741050929194782733001765987770358483, 0.15894419745390376206549481
671071894289; 1, -0.13838372073406036365047976417441696637 + 0.49181637657
768643499753285514741525107*I, -0.22273329410580226599155701611419649154 -
 0.13611876021752805221674918029071012580*I, -0.13167445871785818798769651
537619416009 + 0.13249517760521973840801462296650806543*I, -0.053650958656
997725359297528357602608116 + 0.27622636814169107038138284681568361486*I; 
1, 1.6829412935943127761629561615079976005 + 2.050035122601072617297428698
3598602163*I, -1.3703526062130959637482576769100030014 + 6.900177522288049
4773720769629846373016*I, -8.0696202866361678983472946546849540475 + 8.876
7676785971042450885284301348051602*I, -22.02582114006995415567344987999775
6863 - 8.4306586896999153544710860185447589664*I], [1, 2, 2; -1.0891151457
205048250249527946671612684, -0.27676744146812072730095952834883393274 - 0
.98363275315537286999506571029483050214*I, 3.36588258718862555232591232301
59952011 - 4.1000702452021452345948573967197204327*I; 1.186171800637796459
4796293860483989860, -0.44546658821160453198311403222839298308 + 0.2722375
2043505610443349836058142025160*I, -2.740705212426191927496515353820006002
9 - 13.800355044576098954744153925969274603*I; -0.597410509291947827330017
65987770358483, -0.26334891743571637597539303075238832018 - 0.264990355210
43947681602924593301613087*I, -16.139240573272335796694589309369908095 - 1
7.753535357194208490177056860269610320*I; 0.158944197453903762065494816710
71894289, -0.10730191731399545071859505671520521623 - 0.552452736283382140
76276569363136722973*I, -44.051642280139908311346899759995513726 + 16.8613
17379399830708942172037089517932*I], [5, 2.0000000000000000000000000000000
000000, -2.0000000000000000000000000000000000000, -17.00000000000000000000
0000000000000000, -44.000000000000000000000000000000000000; 2.000000000000
0000000000000000000000000, 15.778109408671998044836357471283695361, 22.314
643349754061651916553814602769764, 10.051395257831478275499932716306366248
, -108.58917507620841447456569092094763671; -2.000000000000000000000000000
0000000000, 22.314643349754061651916553814602769764, 100.52391262388960975
827806174040462368, 143.93295090847353519436673793501057176, -55.842564718
082452641322500190813370023; -17.000000000000000000000000000000000000, 10.
051395257831478275499932716306366248, 143.93295090847353519436673793501057
176, 288.25823756749944693139292174819167135, 205.798400382776623757201806
49465932302; -44.000000000000000000000000000000000000, -108.58917507620841
447456569092094763671, -55.842564718082452641322500190813370023, 205.79840
038277662375720180649465932302, 1112.6092277946777707779250962522343036], 
[5, 2, -2, -17, -44; 2, -2, -34, -63, -40; -2, -34, -90, -101, 177; -17, -
63, -101, -27, 505; -44, -40, 177, 505, 828], [345, 0, 160, 252, 156; 0, 3
45, 215, 311, 306; 0, 0, 5, 3, 2; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [163875, 
-388125, -296700, 234600, -89700; -388125, -1593900, -677925, 595125, -315
675; -296700, -677925, 17250, 58650, -87975; 234600, 595125, 58650, -10005
0, 89700; -89700, -315675, -87975, 89700, -55200], [595125, 0, 357075, 427
800, 326025; 0, 595125, 119025, 512325, 191475; 0, 0, 119025, 79350, 75900
; 0, 0, 0, 1725, 0; 0, 0, 0, 0, 1725]], [-1.089115145720504825024952794667
1612684, -0.13838372073406036365047976417441696637 + 0.4918163765776864349
9753285514741525107*I, 1.6829412935943127761629561615079976005 + 2.0500351
226010726172974286983598602163*I], [1, x, x^2, 1/2*x^3 + 1/2*x^2 + 1/2*x, 
1/2*x^4 + 1/2*x], [1, 0, 0, 0, 0; 0, 1, 0, -1, -1; 0, 0, 1, -1, 0; 0, 0, 0
, 2, 0; 0, 0, 0, 0, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -2, 0
, 0, -1, -2, -2, 0, -1, -2, -2, 5; 0, 1, 0, 0, 0, 1, 0, -1, -1, -1, 0, -1,
 -1, -2, 2, 0, -1, -2, -1, 7, 0, -1, 2, 7, 14; 0, 0, 1, 0, 0, 0, 1, -1, 0,
 -2, 1, -1, 0, -3, -3, 0, 0, -3, -4, -1, 0, -2, -3, -1, 15; 0, 0, 0, 1, 0,
 0, 0, 2, 1, -3, 0, 2, 0, -2, -13, 1, 1, -2, -9, -19, 0, -3, -13, -19, 7; 
0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 1, 3, 4, -4, 1, 2, 1, -4, 
-21]]
? nf3=nfinit(x^6+108);
? nf4=nfinit(x^3-10*x+8)
% = [x^3 - 10*x + 8, [3, 0], 568, 2, [[1, -3.50466435358804770515010852590
43320579, 6.1413361156553641347759399165844441383; 1, 0.864640886695403025
83112842266613688800, 0.37380193147270638662350044992137561317; 1, 2.64002
34668926446793189801032381951699, 3.4848619528719294786005596334941802484]
, [1, 1, 1; -3.5046643535880477051501085259043320579, 0.864640886695403025
83112842266613688800, 2.6400234668926446793189801032381951699; 6.141336115
6553641347759399165844441383, 0.37380193147270638662350044992137561317, 3.
4848619528719294786005596334941802484], [3, -3.454467421397566795000000000
0000000000 E-77, 10.000000000000000000000000000000000000; -3.4544674213975
667950000000000000000000 E-77, 20.000000000000000000000000000000000000, -1
2.000000000000000000000000000000000000; 10.0000000000000000000000000000000
00000, -12.000000000000000000000000000000000000, 50.0000000000000000000000
00000000000000], [3, 0, 10; 0, 20, -12; 10, -12, 50], [284, 168, 235; 0, 2
, 0; 0, 0, 1], [856, -120, -200; -120, 50, 36; -200, 36, 60], [568, 0, 176
; 0, 142, 116; 0, 0, 4]], [-3.5046643535880477051501085259043320579, 0.864
64088669540302583112842266613688800, 2.64002346689264467931898010323819516
99], [1, x, 1/2*x^2], [1, 0, 0; 0, 1, 0; 0, 0, 2], [1, 0, 0, 0, 0, -4, 0, 
-4, 0; 0, 1, 0, 1, 0, 5, 0, 5, -2; 0, 0, 1, 0, 2, 0, 1, 0, 5]]
? 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]), [-2.712465305184343974680
8795106061300699 - 3.1415926535897932384626433832795028842*I; 2.7124653051
843439746808795106061300699 - 6.2831853071795864769252867665590057684*I], 
[-575.96476824756753723069812383847548501 + 3.1415926535897932384626433832
795028842*I, 551.92021890548710036272154926028390729 + 3.14159265358979323
84626433832795028842*I, -2182.8330851707114713996393899372203372 + 9.42477
79607693797153879301498385086526*I, 22773.85870232775201142066437104906806
6 + 6.7262326274067163460000000000000000000 E-44*I, 22749.9470943847253715
22362658767003613 + 8.9683101698756217950000000000000000000 E-44*I, -22485
.248523724199248946630547104902428 + 3.14159265358979323846264338327950288
42*I, 24764.229400428850901332468152110725047 + 4.484155084937810897000000
0000000000000 E-44*I, -0.34328764427702709438988786673341921876 + 3.141592
6535897932384626433832795028842*I, -22509.02540427005566223247521509196639
4 + 3.1415926535897932384626433832795028842*I, 22773.924880629634757152849
739541391231 + 9.4247779607693797153879301498385086526*I; 575.964768247567
53723069812383847548501 + 12.566370614359172953850573533118011536*I, -551.
92021890548710036272154926028390729 + 3.1415926535897932384626433832795028
842*I, 2182.8330851707114713996393899372203372 + 12.5663706143591729538505
73533118011536*I, -22773.858702327752011420664371049068066 + 8.96831016987
56217950000000000000000000 E-44*I, -22749.94709438472537152236265876700361
3 + 6.7262326274067163460000000000000000000 E-44*I, 22485.2485237241992489
46630547104902428 + 9.4247779607693797153879301498385086526*I, -24764.2294
00428850901332468152110725047 + 1.3452465254813432690000000000000000000 E-
43*I, 0.34328764427702709438988786673341921876 + 0.E-48*I, 22509.025404270
055662232475215091966394 + 9.4247779607693797153879301498385086526*I, -227
73.924880629634757152849739541391231 + 8.968310169875621795000000000000000
0000 E-44*I], [[3, [-1, 1]~, 1, 1, [0, 1]~], [3, [0, 1]~, 1, 1, [-1, 1]~],
 [5, [-2, 1]~, 1, 1, [1, 1]~], [5, [1, 1]~, 1, 1, [-2, 1]~], [11, [-2, 1]~
, 1, 1, [1, 1]~], [11, [1, 1]~, 1, 1, [-2, 1]~], [17, [-3, 1]~, 1, 1, [2, 
1]~], [17, [2, 1]~, 1, 1, [-3, 1]~], [19, [-1, 1]~, 1, 1, [0, 1]~], [19, [
0, 1]~, 1, 1, [-1, 1]~]]~, [1, 3, 5, 2, 4, 6, 7, 8, 10, 9]~, [x^2 - x - 57
, [2, 0], 229, 1, [[1, -7.0663729752107779635959310246705326058; 1, 8.0663
729752107779635959310246705326058], [1, 1; -7.0663729752107779635959310246
705326058, 8.0663729752107779635959310246705326058], [2, 1.000000000000000
0000000000000000000000; 1.0000000000000000000000000000000000000, 115.00000
000000000000000000000000000000], [2, 1; 1, 115], [229, 114; 0, 1], [115, -
1; -1, 2], [229, 114; 0, 1]], [-7.0663729752107779635959310246705326058, 8
.0663729752107779635959310246705326058], [1, x], [1, 0; 0, 1], [1, 0, 0, 5
7; 0, 1, 1, 1]], [[3, [3], [[3, 2; 0, 1]]], 2.7124653051843439746808795106
061300699, 0.88144225126545793690341704100000000000, [2, -1], [x + 7], 141
], [Mat([1]), Mat([1]), [[[3, 2; 0, 1], [0.E-48, 0.E-48]]]], 0]
? setrand(1);bnfinit(x^2-x-100000,1)
% = [Mat([5]), Mat([3, 2, 1, 2, 0, 3, 2, 3, 0, 0, 4, 1, 3, 2, 2, 3, 3, 2])
, [-129.82045011403975460991182396195022419 + 6.28318530717958647692528676
65590057684*I; 129.82045011403975460991182396195022419 + 6.283185307179586
4769252867665590057684*I], [-41.811264589129943393339502258694361489 + 12.
566370614359172953850573533118011536*I, 7798.46690725717550557634706376089
75446 + 3.2944368571043014520000000000000000000 E-83*I, 10373.761399242104
962067848600958586774 + 12.566370614359172953850573533118011536*I, -158900
.23093958465964253207252942707441 + 12.56637061435917295385057353311801153
6*I, -325121.45825055946571559418432902593983 + 9.424777960769379715387930
1498385086526*I, -145593.63480289558479501611057332717643 + 3.141592653589
7932384626433832795028842*I, -145558.66123254661945437939684760746451 + 6.
2831853071795864769252867665590057684*I, 184597.94231098623536723383343794
216416 + 9.4247779607693797153879301498385086526*I, -132841.39398007350322
127235517938271907 + 3.1415926535897932384626433832795028842*I, -12674.735
792104327717774025829788414282 + 6.2831853071795864769252867665590057684*I
, -140643.21579257662247653183826927480048 + 6.283185307179586476925286766
5590057684*I, -228948.39559706781480223963544800452232 + 3.141592653589793
2384626433832795028842*I, 26127.041254348588158164026917179577779 + 9.4247
779607693797153879301498385086526*I, -135417.70965704811267121166315150337
705 + 12.566370614359172953850573533118011536*I, -17942.053192221967466487
540449818532080 + 1.1201085313223302360000000000000000000 E-81*I, 7639.699
8076618294564723087059201850803 + 12.566370614359172953850573533118011536*
I, -88300.010593069647012620890888549968664 + 3.14159265358979323846264338
32795028842*I, -90920.002638063436052838364758102886803 + 5.27109897136688
23240000000000000000000 E-82*I, 20904.968762006878374702552140353021602 + 
9.4247779607693797153879301498385086526*I; 41.8112645891299433933395022586
94361489 + 6.2831853071795864769252867665590057684*I, -7798.46690725717550
55763470637608975446 + 9.4247779607693797153879301498385086526*I, -10373.7
61399242104962067848600958586774 + 9.4247779607693797153879301498385086526
*I, 158900.23093958465964253207252942707441 + 12.5663706143591729538505735
33118011536*I, 325121.45825055946571559418432902593983 + 12.56637061435917
2953850573533118011536*I, 145593.63480289558479501611057332717643 + 6.2831
853071795864769252867665590057684*I, 145558.661232546619454379396847607464
51 + 9.4247779607693797153879301498385086526*I, -184597.942310986235367233
83343794216416 + 6.2831853071795864769252867665590057684*I, 132841.3939800
7350322127235517938271907 + 12.566370614359172953850573533118011536*I, 126
74.735792104327717774025829788414282 + 6.588873714208602905000000000000000
0000 E-83*I, 140643.21579257662247653183826927480048 + 6.28318530717958647
69252867665590057684*I, 228948.39559706781480223963544800452232 + 3.141592
6535897932384626433832795028842*I, -26127.04125434858815816402691717957777
9 + 3.1415926535897932384626433832795028842*I, 135417.70965704811267121166
315150337705 + 3.1415926535897932384626433832795028842*I, 17942.0531922219
67466487540449818532080 + 9.4247779607693797153879301498385086526*I, -7639
.6998076618294564723087059201850803 + 3.1415926535897932384626433832795028
842*I, 88300.010593069647012620890888549968664 + 6.28318530717958647692528
67665590057684*I, 90920.002638063436052838364758102886803 + 12.56637061435
9172953850573533118011536*I, -20904.968762006878374702552140353021602 + 3.
1415926535897932384626433832795028842*I], [[2, [1, 1]~, 1, 1, [0, 1]~], [2
, [2, 1]~, 1, 1, [1, 1]~], [5, [4, 1]~, 1, 1, [0, 1]~], [5, [5, 1]~, 1, 1,
 [-1, 1]~], [7, [3, 1]~, 2, 1, [3, 1]~], [13, [-6, 1]~, 1, 1, [5, 1]~], [1
3, [5, 1]~, 1, 1, [-6, 1]~], [17, [14, 1]~, 1, 1, [2, 1]~], [17, [19, 1]~,
 1, 1, [-3, 1]~], [23, [6, 1]~, 1, 1, [-7, 1]~], [23, [-7, 1]~, 1, 1, [6, 
1]~], [29, [13, 1]~, 1, 1, [-14, 1]~], [29, [-14, 1]~, 1, 1, [13, 1]~], [3
1, [23, 1]~, 1, 1, [7, 1]~], [31, [38, 1]~, 1, 1, [-8, 1]~], [41, [-7, 1]~
, 1, 1, [6, 1]~], [41, [6, 1]~, 1, 1, [-7, 1]~], [43, [-16, 1]~, 1, 1, [15
, 1]~], [43, [15, 1]~, 1, 1, [-16, 1]~]]~, [1, 3, 6, 2, 4, 5, 7, 9, 8, 11,
 10, 13, 12, 15, 14, 17, 16, 19, 18]~, [x^2 - x - 100000, [2, 0], 400001, 
1, [[1, -315.72816130129840161392089489603747004; 1, 316.72816130129840161
392089489603747004], [1, 1; -315.72816130129840161392089489603747004, 316.
72816130129840161392089489603747004], [2, 1.000000000000000000000000000000
0000000; 1.0000000000000000000000000000000000000, 200001.00000000000000000
000000000000000], [2, 1; 1, 200001], [400001, 200000; 0, 1], [200001, -1; 
-1, 2], [400001, 200000; 0, 1]], [-315.72816130129840161392089489603747004
, 316.72816130129840161392089489603747004], [1, x], [1, 0; 0, 1], [1, 0, 0
, 100000; 0, 1, 1, 1]], [[5, [5], [[2, 1; 0, 1]]], 129.8204501140397546099
1182396195022419, 0.98765369790690472391212970100000000000, [2, -1], [3795
54884019013781006303254896369154068336082609238336*x + 1198361656442507899
90462835950022871665178127611316131167], 80], [Mat([1]), Mat([1]), [[[2, 1
; 0, 1], [0.E-86, 0.E-86]]]], 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], [-3.233732695981516673, 
-0.07182350902743636344, 4.305556205008953036], [10889, 5698, 3794; 0, 1, 
0; 0, 0, 1], Mat([2]), Mat([1, 0, 1, 1, 0, 1, 1, 1]), [9, 15, 16, 17, 10, 
33, 39, 69, 57], [2, [-1, 0, 0]~], [[0, 1, 0]~, [-4, 2, 1]~], [-4, 2, 3, -
1, 3, 1, -1, 11, -7; 1, 1, 1, 1, 0, 1, -4, 2, -2; 0, 0, 0, 0, 0, 0, -1, -1
, 0]]
? \p38
   realprecision = 38 significant digits
? bnrinit(bnf,[[5,3;0,1],[1,0]],1)
% = [[Mat([3]), Mat([1, 2, 1, 2, 1, 2, 1, 2, 1]), [-2.71246530518434397468
08795106061300699 - 3.1415926535897932384626433832795028842*I; 2.712465305
1843439746808795106061300699 - 6.2831853071795864769252867665590057684*I],
 [-575.96476824756753723069812383847548501 + 3.141592653589793238462643383
2795028842*I, 551.92021890548710036272154926028390729 + 3.1415926535897932
384626433832795028842*I, -2182.8330851707114713996393899372203372 + 9.4247
779607693797153879301498385086526*I, 22773.8587023277520114206643710490680
66 + 6.7262326274067163460000000000000000000 E-44*I, 22749.947094384725371
522362658767003613 + 8.9683101698756217950000000000000000000 E-44*I, -2248
5.248523724199248946630547104902428 + 3.1415926535897932384626433832795028
842*I, 24764.229400428850901332468152110725047 + 4.48415508493781089700000
00000000000000 E-44*I, -0.34328764427702709438988786673341921876 + 3.14159
26535897932384626433832795028842*I, -22509.0254042700556622324752150919663
94 + 3.1415926535897932384626433832795028842*I, 22773.92488062963475715284
9739541391231 + 9.4247779607693797153879301498385086526*I; 575.96476824756
753723069812383847548501 + 12.566370614359172953850573533118011536*I, -551
.92021890548710036272154926028390729 + 3.141592653589793238462643383279502
8842*I, 2182.8330851707114713996393899372203372 + 12.566370614359172953850
573533118011536*I, -22773.858702327752011420664371049068066 + 8.9683101698
756217950000000000000000000 E-44*I, -22749.9470943847253715223626587670036
13 + 6.7262326274067163460000000000000000000 E-44*I, 22485.248523724199248
946630547104902428 + 9.4247779607693797153879301498385086526*I, -24764.229
400428850901332468152110725047 + 1.3452465254813432690000000000000000000 E
-43*I, 0.34328764427702709438988786673341921876 + 0.E-48*I, 22509.02540427
0055662232475215091966394 + 9.4247779607693797153879301498385086526*I, -22
773.924880629634757152849739541391231 + 8.96831016987562179500000000000000
00000 E-44*I], [[3, [-1, 1]~, 1, 1, [0, 1]~], [3, [0, 1]~, 1, 1, [-1, 1]~]
, [5, [-2, 1]~, 1, 1, [1, 1]~], [5, [1, 1]~, 1, 1, [-2, 1]~], [11, [-2, 1]
~, 1, 1, [1, 1]~], [11, [1, 1]~, 1, 1, [-2, 1]~], [17, [-3, 1]~, 1, 1, [2,
 1]~], [17, [2, 1]~, 1, 1, [-3, 1]~], [19, [-1, 1]~, 1, 1, [0, 1]~], [19, 
[0, 1]~, 1, 1, [-1, 1]~]]~, [1, 3, 5, 2, 4, 6, 7, 8, 10, 9]~, [x^2 - x - 5
7, [2, 0], 229, 1, [[1, -7.0663729752107779635959310246705326058; 1, 8.066
3729752107779635959310246705326058], [1, 1; -7.066372975210777963595931024
6705326058, 8.0663729752107779635959310246705326058], [2, 1.00000000000000
00000000000000000000000; 1.0000000000000000000000000000000000000, 115.0000
0000000000000000000000000000000], [2, 1; 1, 115], [229, 114; 0, 1], [115, 
-1; -1, 2], [229, 114; 0, 1]], [-7.0663729752107779635959310246705326058, 
8.0663729752107779635959310246705326058], [1, x], [1, 0; 0, 1], [1, 0, 0, 
57; 0, 1, 1, 1]], [[3, [3], [[3, 2; 0, 1]]], 2.712465305184343974680879510
6061300699, 0.88144225126545793690341704100000000000, [2, -1], [x + 7], 14
1], [Mat([1]), Mat([1]), [[[3, 2; 0, 1], [0.E-48, 0.E-48]]]], 0], [[[5, 3;
 0, 1], [1, 0]], [8, [4, 2], [[2, 0]~, [-4, 0]~]], Mat([[5, [-2, 1]~, 1, 1
, [1, 1]~], 1]), [[[[4], [[2, 0]~], [[2, 0]~], [[Mod(0, 2)]~], 1]], [[2], 
[[-4, 0]~], Mat([1])]], [1, 0; 0, 1]], [[1, 0]~], [1, -3, -6; 0, 0, 1; 0, 
1, 0], [12, [12], [[3, 2; 0, 1]]], [[0, 0; 0, 1], [-1, -1; 1, -1]]]
? bnr=bnrclass(bnf,[[5,3;0,1],[1,0]],2)
% = [[Mat([3]), Mat([1, 2, 1, 2, 1, 2, 1, 2, 1]), [-2.71246530518434397468
08795106061300699 - 3.1415926535897932384626433832795028842*I; 2.712465305
1843439746808795106061300699 - 6.2831853071795864769252867665590057684*I],
 [-575.96476824756753723069812383847548501 + 3.141592653589793238462643383
2795028842*I, 551.92021890548710036272154926028390729 + 3.1415926535897932
384626433832795028842*I, -2182.8330851707114713996393899372203372 + 9.4247
779607693797153879301498385086526*I, 22773.8587023277520114206643710490680
66 + 6.7262326274067163460000000000000000000 E-44*I, 22749.947094384725371
522362658767003613 + 8.9683101698756217950000000000000000000 E-44*I, -2248
5.248523724199248946630547104902428 + 3.1415926535897932384626433832795028
842*I, 24764.229400428850901332468152110725047 + 4.48415508493781089700000
00000000000000 E-44*I, -0.34328764427702709438988786673341921876 + 3.14159
26535897932384626433832795028842*I, -22509.0254042700556622324752150919663
94 + 3.1415926535897932384626433832795028842*I, 22773.92488062963475715284
9739541391231 + 9.4247779607693797153879301498385086526*I; 575.96476824756
753723069812383847548501 + 12.566370614359172953850573533118011536*I, -551
.92021890548710036272154926028390729 + 3.141592653589793238462643383279502
8842*I, 2182.8330851707114713996393899372203372 + 12.566370614359172953850
573533118011536*I, -22773.858702327752011420664371049068066 + 8.9683101698
756217950000000000000000000 E-44*I, -22749.9470943847253715223626587670036
13 + 6.7262326274067163460000000000000000000 E-44*I, 22485.248523724199248
946630547104902428 + 9.4247779607693797153879301498385086526*I, -24764.229
400428850901332468152110725047 + 1.3452465254813432690000000000000000000 E
-43*I, 0.34328764427702709438988786673341921876 + 0.E-48*I, 22509.02540427
0055662232475215091966394 + 9.4247779607693797153879301498385086526*I, -22
773.924880629634757152849739541391231 + 8.96831016987562179500000000000000
00000 E-44*I], [[3, [-1, 1]~, 1, 1, [0, 1]~], [3, [0, 1]~, 1, 1, [-1, 1]~]
, [5, [-2, 1]~, 1, 1, [1, 1]~], [5, [1, 1]~, 1, 1, [-2, 1]~], [11, [-2, 1]
~, 1, 1, [1, 1]~], [11, [1, 1]~, 1, 1, [-2, 1]~], [17, [-3, 1]~, 1, 1, [2,
 1]~], [17, [2, 1]~, 1, 1, [-3, 1]~], [19, [-1, 1]~, 1, 1, [0, 1]~], [19, 
[0, 1]~, 1, 1, [-1, 1]~]]~, [1, 3, 5, 2, 4, 6, 7, 8, 10, 9]~, [x^2 - x - 5
7, [2, 0], 229, 1, [[1, -7.0663729752107779635959310246705326058; 1, 8.066
3729752107779635959310246705326058], [1, 1; -7.066372975210777963595931024
6705326058, 8.0663729752107779635959310246705326058], [2, 1.00000000000000
00000000000000000000000; 1.0000000000000000000000000000000000000, 115.0000
0000000000000000000000000000000], [2, 1; 1, 115], [229, 114; 0, 1], [115, 
-1; -1, 2], [229, 114; 0, 1]], [-7.0663729752107779635959310246705326058, 
8.0663729752107779635959310246705326058], [1, x], [1, 0; 0, 1], [1, 0, 0, 
57; 0, 1, 1, 1]], [[3, [3], [[3, 2; 0, 1]]], 2.712465305184343974680879510
6061300699, 0.88144225126545793690341704100000000000, [2, -1], [x + 7], 14
1], [Mat([1]), Mat([1]), [[[3, 2; 0, 1], [0.E-48, 0.E-48]]]], 0], [[[5, 3;
 0, 1], [1, 0]], [8, [4, 2], [[2, 0]~, [-4, 0]~]], Mat([[5, [-2, 1]~, 1, 1
, [1, 1]~], 1]), [[[[4], [[2, 0]~], [[2, 0]~], [[Mod(0, 2)]~], 1]], [[2], 
[[-4, 0]~], Mat([1])]], [1, 0; 0, 1]], [[1, 0]~], [1, -3, -6; 0, 0, 1; 0, 
1, 0], [12, [12], [[3, 2; 0, 1]]], [[0, 0; 0, 1], [-1, -1; 1, -1]]]
? rnfinit(nf2,x^5-x-2)
% = [x^5 - x - 2, [[1, 2], [0, 5]], [[49744, 0, 0; 0, 49744, 0; 0, 0, 4974
4], [3109, 0, 0]~], [1, 0, 0; 0, 1, 0; 0, 0, 1], [[[1, 1.26716830454212431
72528914279776896412, 1.6057155120361619195949075151301679393, 2.034711802
9638523119874445717108994866, 2.5783223055935536544757871909285592749; 1, 
0.26096388038645528500256735072673484811 + 1.17722615339419443947002865856
17926513*I, -1.3177592693689352747870763902256347904 + 0.61442701016433883
804190660864146731824*I, -1.0672071180669977537495893497477340535 - 1.3909
574189920019216524673160314582604*I, 1.35896894118826157536264394806140019
36 - 1.6193337759893970298359887428575174472*I; 1, -0.89454803265751744362
901306471557966872 + 0.53414854617473272670874609150394379949*I, 0.5149015
1335085431498962263266055082078 - 0.95564306225496055080453352211847466685
*I, 0.049851216585071597755867063892284310224 + 1.129902516042508991899302
4639913611785*I, -0.64813009398503840260053754352567983115 - 0.98412411795
664774269323431620030610541*I], [1, 1.267168304542124317252891427977689641
2 + 0.E-38*I, 1.6057155120361619195949075151301679393 + 0.E-38*I, 2.034711
8029638523119874445717108994866 + 0.E-37*I, 2.5783223055935536544757871909
285592749 + 0.E-37*I; 1, 0.26096388038645528500256735072673484811 - 1.1772
261533941944394700286585617926513*I, -1.3177592693689352747870763902256347
904 - 0.61442701016433883804190660864146731824*I, -1.067207118066997753749
5893497477340535 + 1.3909574189920019216524673160314582604*I, 1.3589689411
882615753626439480614001936 + 1.6193337759893970298359887428575174472*I; 1
, 0.26096388038645528500256735072673484811 + 1.177226153394194439470028658
5617926513*I, -1.3177592693689352747870763902256347904 + 0.614427010164338
83804190660864146731824*I, -1.0672071180669977537495893497477340535 - 1.39
09574189920019216524673160314582604*I, 1.358968941188261575362643948061400
1936 - 1.6193337759893970298359887428575174472*I; 1, -0.894548032657517443
62901306471557966872 - 0.53414854617473272670874609150394379949*I, 0.51490
151335085431498962263266055082078 + 0.955643062254960550804533522118474666
85*I, 0.049851216585071597755867063892284310224 - 1.1299025160425089918993
024639913611785*I, -0.64813009398503840260053754352567983115 + 0.984124117
95664774269323431620030610541*I; 1, -0.89454803265751744362901306471557966
872 + 0.53414854617473272670874609150394379949*I, 0.5149015133508543149896
2263266055082078 - 0.95564306225496055080453352211847466685*I, 0.049851216
585071597755867063892284310224 + 1.1299025160425089918993024639913611785*I
, -0.64813009398503840260053754352567983115 - 0.98412411795664774269323431
620030610541*I]], [[1, 2, 2; 1.2671683045421243172528914279776896412, 0.52
192776077291057000513470145346969622 - 2.354452306788388878940057317123585
3026*I, -1.7890960653150348872580261294311593374 - 1.068297092349465453417
4921830078875989*I; 1.6057155120361619195949075151301679393, -2.6355185387
378705495741527804512695809 - 1.2288540203286776760838132172829346364*I, 1
.0298030267017086299792452653211016415 + 1.9112861245099211016090670442369
493337*I; 2.0347118029638523119874445717108994866, -2.13441423613399550749
91786994954681070 + 2.7819148379840038433049346320629165208*I, 0.099702433
170143195511734127784568620449 - 2.2598050320850179837986049279827223571*I
; 2.5783223055935536544757871909285592749, 2.71793788237652315072528789612
28003872 + 3.2386675519787940596719774857150348944*I, -1.29626018797007680
52010750870513596623 + 1.9682482359132954853864686324006122108*I], [1, 1, 
1, 1, 1; 1.2671683045421243172528914279776896412 + 0.E-38*I, 0.26096388038
645528500256735072673484811 + 1.1772261533941944394700286585617926513*I, 0
.26096388038645528500256735072673484811 - 1.177226153394194439470028658561
7926513*I, -0.89454803265751744362901306471557966872 + 0.53414854617473272
670874609150394379949*I, -0.89454803265751744362901306471557966872 - 0.534
14854617473272670874609150394379949*I; 1.605715512036161919594907515130167
9393 + 0.E-38*I, -1.3177592693689352747870763902256347904 + 0.614427010164
33883804190660864146731824*I, -1.3177592693689352747870763902256347904 - 0
.61442701016433883804190660864146731824*I, 0.51490151335085431498962263266
055082078 - 0.95564306225496055080453352211847466685*I, 0.5149015133508543
1498962263266055082078 + 0.95564306225496055080453352211847466685*I; 2.034
7118029638523119874445717108994866 + 0.E-37*I, -1.067207118066997753749589
3497477340535 - 1.3909574189920019216524673160314582604*I, -1.067207118066
9977537495893497477340535 + 1.3909574189920019216524673160314582604*I, 0.0
49851216585071597755867063892284310224 + 1.1299025160425089918993024639913
611785*I, 0.049851216585071597755867063892284310224 - 1.129902516042508991
8993024639913611785*I; 2.5783223055935536544757871909285592749 + 0.E-37*I,
 1.3589689411882615753626439480614001936 - 1.61933377598939702983598874285
75174472*I, 1.3589689411882615753626439480614001936 + 1.619333775989397029
8359887428575174472*I, -0.64813009398503840260053754352567983115 - 0.98412
411795664774269323431620030610541*I, -0.6481300939850384026005375435256798
3115 + 0.98412411795664774269323431620030610541*I]], [[5, -5.8774717524647
712700000000000000000000 E-39 + 3.4227493991378543323575495001314729016*I,
 2.3509887009859085080000000000000000000 E-38 - 0.682432104181243425525253
82695401469720*I, -2.3509887009859085080000000000000000000 E-38 - 0.522109
80589898585950632970408019416371*I, 3.999999999999999999999999999999999999
9 - 5.2069157878920895450584461181156471052*I; -5.877471752464771270000000
0000000000000 E-39 - 3.4227493991378543323575495001314729016*I, 6.68470434
24634879841147654217963674264 - 5.8774717524647712700000000000000000000 E-
39*I, 0.85145677340721376574333983502938573598 + 4.58295731809784302915415
92600601794652*I, -0.13574266252716976137461193821267520737 - 0.2880510854
4025772361738936467682050391*I, 0.27203784387468568916539788233281013320 -
 1.5917147279942947718965650859986677247*I; 2.3509887009859085080000000000
000000000 E-38 + 0.68243210418124342552525382695401469720*I, 0.85145677340
721376574333983502938573598 - 4.5829573180978430291541592600601794652*I, 9
.1630968530221077951281598310681467898 + 0.E-38*I, 2.262298765209562945340
3849736225691490 + 6.2361927913558506765724047063180706869*I, -0.217964098
86496632254445901043974770643 + 0.34559368931063215686158939748833975810*I
; -2.3509887009859085080000000000000000000 E-38 + 0.5221098058989858595063
2970408019416371*I, -0.13574266252716976137461193821267520737 + 0.28805108
544025772361738936467682050392*I, 2.2622987652095629453403849736225691490 
- 6.2361927913558506765724047063180706869*I, 12.84576894883233551188269693
9380696155 + 1.1754943504929542540000000000000000000 E-38*I, 4.56184005023
78124720913214622468855074 + 8.6033930051068500425218923146793019614*I; 3.
9999999999999999999999999999999999999 + 5.20691578789208954505844611811564
71052*I, 0.27203784387468568916539788233281013320 + 1.59171472799429477189
65650859986677247*I, -0.21796409886496632254445901043974770643 - 0.3455936
8931063215686158939748833975810*I, 4.5618400502378124720913214622468855074
 - 8.6033930051068500425218923146793019615*I, 18.3629686304161144024252991
86062892646 + 5.8774717524647712700000000000000000000 E-39*I], [5, -1.1754
943504929542540000000000000000000 E-38 + 0.E-38*I, 2.350988700985908508000
0000000000000000 E-38 + 0.E-38*I, -1.7632415262050926680000000000000000000
 E-38 + 0.E-38*I, 3.9999999999999999999999999999999999998 + 0.E-38*I; -1.1
754943504929542540000000000000000000 E-38 + 0.E-38*I, 6.684704342463487984
1147654217963674264 - 5.8774717524647712700000000000000000000 E-39*I, 0.85
145677340721376574333983502938573597 + 5.877471752464771270000000000000000
0000 E-39*I, -0.13574266252716976137461193821267520737 + 5.877471752464771
2700000000000000000000 E-39*I, 0.27203784387468568916539788233281013314 - 
5.8774717524647712700000000000000000000 E-39*I; 2.350988700985908508000000
0000000000000 E-38 + 0.E-38*I, 0.85145677340721376574333983502938573597 + 
5.8774717524647712700000000000000000000 E-39*I, 9.163096853022107795128159
8310681467898 + 0.E-38*I, 2.2622987652095629453403849736225691490 + 2.3509
887009859085080000000000000000000 E-38*I, -0.21796409886496632254445901043
974770651 + 0.E-38*I; -1.7632415262050926680000000000000000000 E-38 + 0.E-
38*I, -0.13574266252716976137461193821267520737 + 5.8774717524647712700000
000000000000000 E-39*I, 2.2622987652095629453403849736225691490 + 2.350988
7009859085080000000000000000000 E-38*I, 12.8457689488323355118826969393806
96155 + 0.E-37*I, 4.5618400502378124720913214622468855073 - 3.526483052410
1853370000000000000000000 E-38*I; 3.9999999999999999999999999999999999998 
+ 0.E-38*I, 0.27203784387468568916539788233281013314 - 5.87747175246477127
00000000000000000000 E-39*I, -0.21796409886496632254445901043974770651 + 0
.E-38*I, 4.5618400502378124720913214622468855073 - 3.526483052410185337000
0000000000000000 E-38*I, 18.362968630416114402425299186062892646 + 0.E-37*
I]], [Mod(5, y^3 - y - 1), 0, 0, 0, Mod(4, y^3 - y - 1); 0, 0, 0, Mod(4, y
^3 - y - 1), Mod(10, y^3 - y - 1); 0, 0, Mod(4, y^3 - y - 1), Mod(10, y^3 
- y - 1), 0; 0, Mod(4, y^3 - y - 1), Mod(10, y^3 - y - 1), 0, 0; Mod(4, y^
3 - y - 1), Mod(10, y^3 - y - 1), 0, 0, Mod(4, y^3 - y - 1)], [;], [;], [;
]], [[1.2671683045421243172528914279776896412, 0.2609638803864552850025673
5072673484811 + 1.1772261533941944394700286585617926513*I, -0.894548032657
51744362901306471557966872 + 0.53414854617473272670874609150394379949*I], 
[1.2671683045421243172528914279776896412 + 0.E-38*I, 0.2609638803864552850
0256735072673484811 - 1.1772261533941944394700286585617926513*I, 0.2609638
8038645528500256735072673484811 + 1.1772261533941944394700286585617926513*
I, -0.89454803265751744362901306471557966872 - 0.5341485461747327267087460
9150394379949*I, -0.89454803265751744362901306471557966872 + 0.53414854617
473272670874609150394379949*I]~], [[Mod(1, y^3 - y - 1), Mod(1, y^3 - y - 
1)*x, Mod(1, y^3 - y - 1)*x^2, Mod(1, y^3 - y - 1)*x^3, Mod(1, y^3 - y - 1
)*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]]], [Mod(1, y^3 - y - 1), 0, 0, 0, 0; 0, Mod(1, y^3 - y - 1), 0, 0, 0;
 0, 0, Mod(1, y^3 - y - 1), 0, 0; 0, 0, 0, Mod(1, y^3 - y - 1), 0; 0, 0, 0
, 0, Mod(1, y^3 - y - 1)], [], [y^3 - y - 1, [1, 1], -23, 1, [[1, 1.324717
9572447460259609088544780973407, 1.7548776662466927600495088963585286918; 
1, -0.66235897862237301298045442723904867036 + 0.5622795120623012438991821
4490937306149*I, 0.12256116687665361997524555182073565405 - 0.744861766619
74423659317042860439236724*I], [1, 2; 1.3247179572447460259609088544780973
407, -1.3247179572447460259609088544780973407 - 1.124559024124602487798364
2898187461229*I; 1.7548776662466927600495088963585286918, 0.24512233375330
723995049110364147130810 + 1.4897235332394884731863408572087847344*I], [3,
 0.E-96, 2.0000000000000000000000000000000000000; 0.E-96, 3.26463299874007
82801485266890755860756, 1.3247179572447460259609088544780973407; 2.000000
0000000000000000000000000000000, 1.3247179572447460259609088544780973407, 
4.2192762054875453178332176670757633303], [3, 0, 2; 0, 2, 3; 2, 3, 2], [23
, 13, 15; 0, 1, 0; 0, 0, 1], [-5, 6, -4; 6, 2, -9; -4, -9, 6], [23, 0, 7; 
0, 23, 10; 0, 0, 1]], [1.3247179572447460259609088544780973407, -0.6623589
7862237301298045442723904867036 + 0.56227951206230124389918214490937306149
*I], [1, y, y^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 1, 0, 1, 0;
 0, 1, 0, 1, 0, 1, 0, 1, 1; 0, 0, 1, 0, 1, 0, 1, 0, 1]], [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(39516536165538345/837185878794
73471*x^14 - 6500512476832995/83718587879473471*x^13 - 196215472046117185/
83718587879473471*x^12 + 229902227480108910/83718587879473471*x^11 + 23738
0704030959181/83718587879473471*x^10 - 1064931988160773805/837185878794734
71*x^9 - 20657086671714300/83718587879473471*x^8 + 1772885205999206010/837
18587879473471*x^7 + 5952033217241102348/83718587879473471*x^6 - 483884018
7320655696/83718587879473471*x^5 + 5180390720553188700/83718587879473471*x
^4 - 8374015687535120430/83718587879473471*x^3 + 8907744727915040221/83718
587879473471*x^2 + 4155976664123434381/83718587879473471*x + 3189202157185
80450/83718587879473471, 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 + 25*
x + 1), -1, [1, x, x^2, x^3, x^4, x^5, x^6, x^7, x^8, x^9, x^10, x^11, x^1
2, x^13, 1/83718587879473471*x^14 - 20528463024680133/83718587879473471*x^
13 - 4742392948888610/83718587879473471*x^12 - 9983523646123358/8371858787
9473471*x^11 + 40898955597139011/83718587879473471*x^10 + 2941269242397193
7/83718587879473471*x^9 - 5017479463612351/83718587879473471*x^8 + 4101499
3230075066/83718587879473471*x^7 - 2712810874903165/83718587879473471*x^6 
+ 20152905879672878/83718587879473471*x^5 + 9591643151927789/8371858787947
3471*x^4 - 8471905745957397/83718587879473471*x^3 - 13395753879413605/8371
8587879473471*x^2 + 27623037732247492/83718587879473471*x + 26306699661480
593/83718587879473471], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -263066
99661480593; 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -27623037732247492;
 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13395753879413605; 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8471905745957397; 0, 0, 0, 0, 1, 0, 0, 0, 0,
 0, 0, 0, 0, 0, -9591643151927789; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, -20152905879672878; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 271281087
4903165; 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -41014993230075066; 0, 
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5017479463612351; 0, 0, 0, 0, 0, 0,
 0, 0, 0, 1, 0, 0, 0, 0, -29412692423971937; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 1, 0, 0, 0, -40898955597139011; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
 9983523646123358; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4742392948888
610; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20528463024680133; 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 83718587879473471]]]
? ? 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], [[2, 1, 1, 1; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]]]

[14.229975145405511722395637833443108790]

[1.1211171071527562299744232290000000000]

? setrand(1);bnfclassunit(x^2-x-100000)
  ***   Warning: insufficient precision for fundamental units, not given.
% = 
[x^2 - x - 100000]

[[2, 0]]

[[400001, 1]]

[[1, x]]

[[5, [5], [[2, 1; 0, 1]]]]

[129.82045011403975460991182396195022419]

[0.98765369790690472391212970100000000000]

[[2, -1]]

[[;]]

[0]

? setrand(1);bnfclassunit(x^2-x-100000,1)
% = 
[x^2 - x - 100000]

[[2, 0]]

[[400001, 1]]

[[1, x]]

[[5, [5], [[2, 1; 0, 1]]]]

[129.82045011403975460991182396195022419]

[0.98765369790690472391212970100000000000]

[[2, -1]]

[[379554884019013781006303254896369154068336082609238336*x + 1198361656442
50789990462835950022871665178127611316131167]]

[80]

? 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, x, 1/3*x^2, 1/1029*x^3 + 33/343*x^2 - 155/343*x - 58/343]]

[[4, [4], [[7, 5, 1, 0; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]]]

[3.7941269688216589341408274220859400302]

[0.88260182866555813061644128400000000000]

[[6, 10/1029*x^3 - 13/343*x^2 + 165/343*x + 1478/343]]

[[4/1029*x^3 + 53/1029*x^2 + 66/343*x + 111/343]]

[137]

? 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, 4, 4, 3; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]]
? bnrconductor(bnf,[[25,13;0,1],[1,1]])
% = [[5, 3; 0, 1], [1, 0]]
? bnrconductorofchar(bnr,[2])
% = [[5, 3; 0, 1], [0, 0]]
? ? ? ? bnfisprincipal(bnf,[5,1;0,1],0)
% = [1]~
? bnfisprincipal(bnf,[5,1;0,1])
% = [[1]~, [2, 1/3]~, 129]
? ? 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, 0, 1, 1, 0, 1, 1, 1]), [1.173637103435061715 + 3.14
1592653589793238*I, -4.562279014988837901 + 3.141592653589793238*I; -2.633
543432738976049 + 3.141592653589793238*I, 1.420330600779487358 + 3.1415926
53589793238*I; 1.459906329303914334, 3.141948414209350543], [1.24634698933
4819161 + 3.141592653589793238*I, -0.6926391142471042845 + 3.1415926535897
93238*I, -1.990056445584799713 + 3.141592653589793238*I, 0.540400637612946
9727 + 3.141592653589793238*I, 0.E-96, 0.004375616572659815402 + 3.1415926
53589793238*I, -0.8305625946607188639, 0.3677262014027817705 + 3.141592653
589793238*I, -1.977791147836553953 + 3.141592653589793238*I; 0.67168274328
67392935 + 3.141592653589793238*I, -0.2461086674077943078, 0.5379005671092
853266, -0.8333219883742404172 + 3.141592653589793238*I, 0.E-96, -0.873831
8043071131265, -1.552661549868775853 + 3.141592653589793238*I, 0.972906318
8316092378, 0.5774919091398324092 + 3.141592653589793238*I; -1.91802973262
1558454, 0.9387477816548985923, 1.452155878475514386, 0.292921350761293444
4, 0.E-96, 0.8694561877344533111, 2.383224144529494717 + 3.141592653589793
238*I, -1.340632520234391008, 1.400299238696721544 + 3.141592653589793238*
I], [[3, [-1, 1, 0]~, 1, 1, [1, 0, 1]~], [5, [2, 1, 0]~, 1, 1, [2, 2, 1]~]
, [5, [3, 1, 0]~, 1, 1, [-2, 1, 1]~], [5, [-1, 1, 0]~, 1, 1, [1, 0, 1]~], 
[3, [1, 0, 1]~, 1, 2, [-1, 1, 0]~], [11, [1, 1, 0]~, 1, 1, [-1, -2, 1]~], 
[13, [19, 1, 0]~, 1, 1, [2, 6, 1]~], [23, [-10, 1, 0]~, 1, 1, [7, 9, 1]~],
 [19, [-6, 1, 0]~, 1, 1, [-3, 5, 1]~]]~, [1, 2, 3, 4, 5, 6, 7, 8, 9]~, [x^
3 - x^2 - 14*x - 1, [3, 0], 10889, 1, [[1, -3.233732695981516673, 10.45702
714905988813; 1, -0.07182350902743636344, 0.005158616449014232794; 1, 4.30
5556205008953036, 18.53781423449109762], [1, 1, 1; -3.233732695981516673, 
-0.07182350902743636344, 4.305556205008953036; 10.45702714905988813, 0.005
158616449014232794, 18.53781423449109762], [3, 1.000000000000000000, 29.00
000000000000000; 1.000000000000000000, 29.00000000000000000, 46.0000000000
0000000; 29.00000000000000000, 46.00000000000000000, 453.0000000000000000]
, [3, 1, 29; 1, 29, 46; 29, 46, 453], [10889, 5698, 3794; 0, 1, 0; 0, 0, 1
], [11021, 881, -795; 881, 518, -109; -795, -109, 86], [10889, 0, 1890; 0,
 10889, 5190; 0, 0, 1]], [-3.233732695981516673, -0.07182350902743636344, 
4.305556205008953036], [1, x, x^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0,
 0, 0, 1, 0, 1, 1; 0, 1, 0, 1, 0, 14, 0, 14, 15; 0, 0, 1, 0, 1, 1, 1, 1, 1
5]], [[2, [2], [[3, 2, 2; 0, 1, 0; 0, 0, 1]]], 10.34800724602767998, 1.000
000000000000000, [2, -1], [x, x^2 + 2*x - 4], 1000], [Mat([1]), Mat([1]), 
[[[3, 2, 2; 0, 1, 0; 0, 0, 1], [0.E-96, 0.E-96, 0.E-96]]]], 0]
? \p38
   realprecision = 38 significant digits
? bnfnarrow(bnf)
% = [3, [3], [[3, 2; 0, 1]]]
? bnfreg(x^2-x-57)
% = 2.7124653051843439746808795106061300699
? bnfsignunit(bnf)
% = 
[-1]

[1]

? ? bnfunit(bnf)
% = [[x + 7], 141]
? bnrclass(bnf,[[5,3;0,1],[1,0]])
% = [12, [12], [[3, 2; 0, 1]]]
? bnr2=bnrclass(bnf,[[25,13;0,1],[1,1]],2)
% = [[Mat([3]), Mat([1, 2, 1, 2, 1, 2, 1, 2, 1]), [-2.71246530518434397468
08795106061300699 - 3.1415926535897932384626433832795028842*I; 2.712465305
1843439746808795106061300699 - 6.2831853071795864769252867665590057684*I],
 [-575.96476824756753723069812383847548501 + 3.141592653589793238462643383
2795028842*I, 551.92021890548710036272154926028390729 + 3.1415926535897932
384626433832795028842*I, -2182.8330851707114713996393899372203372 + 9.4247
779607693797153879301498385086526*I, 22773.8587023277520114206643710490680
66 + 6.7262326274067163460000000000000000000 E-44*I, 22749.947094384725371
522362658767003613 + 8.9683101698756217950000000000000000000 E-44*I, -2248
5.248523724199248946630547104902428 + 3.1415926535897932384626433832795028
842*I, 24764.229400428850901332468152110725047 + 4.48415508493781089700000
00000000000000 E-44*I, -0.34328764427702709438988786673341921876 + 3.14159
26535897932384626433832795028842*I, -22509.0254042700556622324752150919663
94 + 3.1415926535897932384626433832795028842*I, 22773.92488062963475715284
9739541391231 + 9.4247779607693797153879301498385086526*I; 575.96476824756
753723069812383847548501 + 12.566370614359172953850573533118011536*I, -551
.92021890548710036272154926028390729 + 3.141592653589793238462643383279502
8842*I, 2182.8330851707114713996393899372203372 + 12.566370614359172953850
573533118011536*I, -22773.858702327752011420664371049068066 + 8.9683101698
756217950000000000000000000 E-44*I, -22749.9470943847253715223626587670036
13 + 6.7262326274067163460000000000000000000 E-44*I, 22485.248523724199248
946630547104902428 + 9.4247779607693797153879301498385086526*I, -24764.229
400428850901332468152110725047 + 1.3452465254813432690000000000000000000 E
-43*I, 0.34328764427702709438988786673341921876 + 0.E-48*I, 22509.02540427
0055662232475215091966394 + 9.4247779607693797153879301498385086526*I, -22
773.924880629634757152849739541391231 + 8.96831016987562179500000000000000
00000 E-44*I], [[3, [-1, 1]~, 1, 1, [0, 1]~], [3, [0, 1]~, 1, 1, [-1, 1]~]
, [5, [-2, 1]~, 1, 1, [1, 1]~], [5, [1, 1]~, 1, 1, [-2, 1]~], [11, [-2, 1]
~, 1, 1, [1, 1]~], [11, [1, 1]~, 1, 1, [-2, 1]~], [17, [-3, 1]~, 1, 1, [2,
 1]~], [17, [2, 1]~, 1, 1, [-3, 1]~], [19, [-1, 1]~, 1, 1, [0, 1]~], [19, 
[0, 1]~, 1, 1, [-1, 1]~]]~, [1, 3, 5, 2, 4, 6, 7, 8, 10, 9]~, [x^2 - x - 5
7, [2, 0], 229, 1, [[1, -7.0663729752107779635959310246705326058; 1, 8.066
3729752107779635959310246705326058], [1, 1; -7.066372975210777963595931024
6705326058, 8.0663729752107779635959310246705326058], [2, 1.00000000000000
00000000000000000000000; 1.0000000000000000000000000000000000000, 115.0000
0000000000000000000000000000000], [2, 1; 1, 115], [229, 114; 0, 1], [115, 
-1; -1, 2], [229, 114; 0, 1]], [-7.0663729752107779635959310246705326058, 
8.0663729752107779635959310246705326058], [1, x], [1, 0; 0, 1], [1, 0, 0, 
57; 0, 1, 1, 1]], [[3, [3], [[3, 2; 0, 1]]], 2.712465305184343974680879510
6061300699, 0.88144225126545793690341704100000000000, [2, -1], [x + 7], 14
1], [Mat([1]), Mat([1]), [[[3, 2; 0, 1], [0.E-48, 0.E-48]]]], 0], [[[25, 1
3; 0, 1], [1, 1]], [80, [20, 2, 2], [[2, 0]~, [11, -5]~, [15, 3]~]], Mat([
[5, [-2, 1]~, 1, 1, [1, 1]~], 2]), [[[[4], [[2, 0]~], [[2, 0]~], [[Mod(0, 
2), Mod(0, 2)]~], 1], [[5], [[6, 0]~], [[6, 0]~], [[Mod(0, 2), Mod(0, 2)]~
], Mat([1/5, -13/5])]], [[2, 2], [[11, -5]~, [15, 3]~], [0, 1; 1, 0]]], [1
, -12, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]], [[1, 0]~], [1, -3, 0, -6; 0, 0, 1, 
0; 0, 0, 0, 1; 0, 1, 0, 0], [12, [12], [[3, 2; 0, 1]]], [[1/2, 5, -9; -1/2
, -5, 10], [-2, 0; 0, 10]]]
? bnrclassno(bnf,[[5,3;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, 12, 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, 13; 0, 1]]
? bnrdisc(bnr,,,1)
% = [12, 1, [1953125, 1160888; 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, 12]]], [[]], [[], []], [], [[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]], [12, 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, 12, [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,,1)
% = [[[[matrix(0,2,j,k,0), [[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], [0, 0, 0], [12, 6, [-1, 1; 
3, 3; 229, 6]], [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]], [0, 0, 0], [24, 12, [5, 9; 229, 12]], [0, 0, 0]]]], [], [], [], [[M
at([12, 2]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]], [[12, 1; 13, 1
], [[0, 0, 0], [12, 6, [-1, 1; 3, 6; 229, 6]], [0, 0, 0], [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], [0, 0, 0], [12, 6, [-1, 1; 11, 3; 229, 6
]], [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, 24]]]], [[13, 1; 20, 1], [[0, 0, 0]
, [24, 12, [3, 6; 5, 9; 229, 12]], [24, 12, [3, 6; 5, 6; 229, 12]], [48, 0
, [3, 12; 5, 18; 229, 24]]]], [[12, 1; 21, 1], [[0, 0, 0], [24, 12, [3, 6;
 5, 9; 229, 12]], [0, 0, 0], [48, 0, [3, 12; 5, 18; 229, 24]]]], [[13, 1; 
21, 1], [[24, 24, [3, 6; 5, 9; 229, 12]], [0, 0, 0], [48, 24, [3, 12; 5, 1
8; 229, 24]], [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], [12, 6, [-1, 1; 17, 3; 229, 6]], [0, 0, 0]
, [0, 0, 0]]], [Mat([69, 1]), [[0, 0, 0], [12, 6, [-1, 1; 17, 3; 229, 6]],
 [0, 0, 0], [0, 0, 0]]]], [], [[Mat([76, 1]), [[18, 18, [19, 6; 229, 9]], 
[0, 0, 0], [0, 0, 0], [36, 0, [19, 15; 229, 18]]]], [Mat([77, 1]), [[18, 1
8, [19, 6; 229, 9]], [0, 0, 0], [36, 18, [-1, 1; 19, 15; 229, 18]], [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]~, [329/6561, -7/19683]~, 125]
? 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, 1, 0, 0, 0]~, 1, 1, [1, -1, -1, 0, 0]~]
? idx=idealmul(nf,matid(5),vp)
% = 
[3 1 2 2 2]

[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 2/3 0 0]

[0 1 1/3 0 0]

[0 0 1/3 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? idy=idealred(nf,idx,[1,5,6])
% = 
[5 0 0 2 0]

[0 5 0 0 0]

[0 0 5 2 0]

[0 0 0 1 0]

[0 0 0 0 5]

? idx2=idealmul(nf,idx,idx)
% = 
[9 7 5 8 2]

[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 1]

[0 2 0 0 1]

[0 0 2 0 0]

[0 0 0 2 1]

[0 0 0 0 1]

? idz=idealintersect(nf,idx,idy)
% = 
[15 5 10 12 10]

[0 5 0 0 0]

[0 0 5 2 0]

[0 0 0 1 0]

[0 0 0 0 5]

? aid=[idx,idy,idz,matid(5),idx]
% = [[3, 1, 2, 2, 2; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0,
 0, 1], [5, 0, 0, 2, 0; 0, 5, 0, 0, 0; 0, 0, 5, 2, 0; 0, 0, 0, 1, 0; 0, 0,
 0, 0, 5], [15, 5, 10, 12, 10; 0, 5, 0, 0, 0; 0, 0, 5, 2, 0; 0, 0, 0, 1, 0
; 0, 0, 0, 0, 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], [3, 1, 2, 2, 2; 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], [[0, 1, 0]~], [[-26, -27, 0]~], [[]~], 1]], [[[26], [[0, 2, 0]~], [[
-27, 2, 0]~], [[]~], 1], [[3, 3, 3], [[1, 3, 0]~, [1, 0, 3]~, [4, 0, 0]~],
 [[1, -24, 0]~, [1, 0, -24]~, [-23, 0, 0]~], [[]~, []~, []~], [0, 1/3, 0; 
0, 0, 1/3; 1/3, 0, 0]], [[3, 3, 3], [[1, 9, 0]~, [1, 0, 9]~, [10, 0, 0]~],
 [[1, -18, 0]~, [1, 0, -18]~, [-17, 0, 0]~], [[]~, []~, []~], [0, 1/9, 0; 
0, 0, 1/9; 1/9, 0, 0]]], [[], [], [;]]], [468, 469, 0, 0, -48776, 0, 0, -3
6582; 0, 0, 1, 0, -7, -6, 0, -3; 0, 0, 0, 1, -3, 0, -6, 0]]
? vaid=[idx,idy,matid(5)]
% = [[3, 1, 2, 2, 2; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0,
 0, 1], [5, 0, 0, 2, 0; 0, 5, 0, 0, 0; 0, 0, 5, 2, 0; 0, 0, 0, 1, 0; 0, 0,
 0, 0, 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]]
? 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, 0, 2, 1, 0]~, [-2, 0, -2, -1, 0]~]
? idealaddtoone(nf,[idy,idx])
% = [[-5, 0, 0, 0, 0]~, [6, 0, 0, 0, 0]~]
? idealappr(nf,idy)
% = [-2, 0, -2, 4, 0]~
? idealappr(nf,idealfactor(nf,idy),1)
% = [-2, 0, -2, 4, 0]~
? ? idealcoprime(nf,idx,idx)
% = [7/3, 2/3, -1/3, -1, 0]~
? idealdiv(nf,idy,idt)
% = 
[5 5/2 5/2 7/2 0]

[0 5/2 0 0 0]

[0 0 5/2 1 0]

[0 0 0 1/2 0]

[0 0 0 0 5/2]

? idealdiv(nf,idx2,idx,1)
% = 
[3 1 2 2 2]

[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, 1, 0, 0, 0]~, 1, 1, [1, -1, -1, 0, 0]~] 1]

[[5, [0, 0, 0, 0, 1]~, 4, 1, [2, 5, 2, 1, 0]~] 3]

[[5, [-2, 0, 0, 0, 1]~, 1, 1, [2, 2, 1, 1, 4]~] 1]

? idealhnf(nf,vp)
% = 
[3 1 2 2 2]

[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 1 2 2 2]

[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, 2; 0, 1], [3, 0; 0, 1]], [[2, 0; 0, 2]], [[5
, 3; 0, 1], [5, 1; 0, 1]], [], [], [], [[9, 5; 0, 1], [3, 0; 0, 3], [9, 3;
 0, 1]], [], [[11, 9; 0, 1], [11, 1; 0, 1]], [[6, 4; 0, 2], [6, 0; 0, 2]],
 [], [], [[15, 8; 0, 1], [15, 3; 0, 1], [15, 11; 0, 1], [15, 6; 0, 1]], [[
4, 0; 0, 4]], [[17, 14; 0, 1], [17, 2; 0, 1]], [], [[19, 18; 0, 1], [19, 0
; 0, 1]], [[10, 6; 0, 2], [10, 2; 0, 2]]]
? ? ideallog(nf2,w,bid)
% = [1574, 8, 6]~
? idealmin(nf,idx,[1,2,3,4,5])
% = [[-1; 0; 0; 1; 0], [2.0885812311199768913287869744681966008 + 3.141592
6535897932384626433832795028842*I, 1.5921096812520196555597562531657929785
 + 4.2447196639216499665715751642189271112*I, -0.7903191544758318546808206
3233076160203 + 2.5437460822678889883600220330800078854*I]]
? idealnorm(nf,idt)
% = 16
? idp=idealpow(nf,idx,7)
% = 
[2187 1807 2129 692 1379]

[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)
% = 
[2 0 0 0 1]

[0 2 0 0 1]

[0 0 2 0 0]

[0 0 0 2 1]

[0 0 0 0 1]

? idealprimedec(nf,2)
% = [[2, [3, 1, 0, 0, 0]~, 1, 1, [1, 1, 0, 1, 1]~], [2, [-3, -5, -4, 3, 15
]~, 1, 4, [1, 1, 0, 0, 0]~]]
? idealprimedec(nf,3)
% = [[3, [1, 1, 0, 0, 0]~, 1, 1, [1, -1, -1, 0, 0]~], [3, [0, 0, 1, 1, 0]~
, 2, 2, [2, 1, 3, 2, 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]

[0]

[1]

[3]

[0]

? idealtwoelt(nf,idy)
% = [5, [2, 0, 2, 1, 0]~]
? idealtwoelt(nf,idy,10)
% = [-2, 0, -2, -1, 0]~
? 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, 1, 0]~], [[-26, -27, 0]~], [[]~], 1]], [[[26], [[0, 2, 0]~], [[
-27, 2, 0]~], [[]~], 1], [[3, 3, 3], [[1, 3, 0]~, [1, 0, 3]~, [4, 0, 0]~],
 [[1, -24, 0]~, [1, 0, -24]~, [-23, 0, 0]~], [[]~, []~, []~], [0, 1/3, 0; 
0, 0, 1/3; 1/3, 0, 0]], [[3, 3, 3], [[1, 9, 0]~, [1, 0, 9]~, [10, 0, 0]~],
 [[1, -18, 0]~, [1, 0, -18]~, [-17, 0, 0]~], [[]~, []~, []~], [0, 1/9, 0; 
0, 0, 1/9; 1/9, 0, 0]]], [[], [], [;]]], [468, 469, 0, 0, -48776, 0, 0, -3
6582; 0, 0, 1, 0, -7, -6, 0, -3; 0, 0, 0, 1, -3, 0, -6, 0]]
? idealval(nf,idp,vp)
% = 7
? ideleprincipal(nf,Mod(x^3+5,nfpol))
% = [[6; 0; 1; 3; 0], [2.2324480827796254080981385584384939684 + 3.1415926
535897932384626433832795028842*I, 5.0387659675158716386435353106610489968 
+ 1.5851760343512250049897278861965702423*I, 4.266404027265102874362591079
7589683173 - 0.0083630478144368246110910258645462996191*I]]
? ? ba=nfalgtobasis(nf,Mod(x^3+5,nfpol))
% = [6, 0, 1, 3, 0]~
? bb=nfalgtobasis(nf,Mod(x^3+x,nfpol))
% = [1, 1, 1, 3, 0]~
? bc=matalgtobasis(nf,[Mod(x^2+x,nfpol);Mod(x^2+1,nfpol)])
% = 
[[0, 1, 1, 0, 0]~]

[[1, 0, 1, 0, 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
/139623738889203638909659*x^4 - 1552451622081122020/1396237388892036389096
59*x^3 + 418509858130821123141/139623738889203638909659*x^2 - 681091379850
75994073134/139623738889203638909659*x - 13185339461968406/583468089969204
47]
? da=nfdetint(nf,[a,aid])
% = 
[30 5 25 27 10]

[0 5 0 0 0]

[0 0 5 2 0]

[0 0 0 1 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)
% = [755/373, -152/373, 159/373, 120/373, -264/373]~
? nfeltdiveuc(nf,ba,bb)
% = [2, 0, 0, 0, -1]~
? nfeltdivrem(nf,ba,bb)
% = [[2, 0, 0, 0, -1]~, [-12, -7, 0, 9, 5]~]
? nfeltmod(nf,ba,bb)
% = [-12, -7, 0, 9, 5]~
? nfeltmul(nf,ba,bb)
% = [-25, -50, -30, 15, 90]~
? nfeltpow(nf,bb,5)
% = [23455, 156370, 115855, 74190, -294375]~
? nfeltreduce(nf,ba,idx)
% = [1, 0, 0, 0, 0]~
? nfeltval(nf,ba,vp)
% = 0
? nffactor(nf2,x^3+x)
% = 
[Mod(1, y^3 - y - 1)*x 1]

[Mod(1, y^3 - y - 1)*x^2 + Mod(1, y^3 - y - 1) 1]

? ? nfgaloisconj(nf3)
% = [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, -x]
? aut=nfgaloisconj(nf3,1)
% = [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, -x]
? nfgaloisapply(nf3,aut[2],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]~, [0, 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]~], [[2, 1, 1, 1, 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], [3, 1, 2, 2, 2; 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]~, [0, 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]~], [[2, 1, 1, 1, 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], [3, 1, 2, 2, 2; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 
0, 0, 0, 1, 0; 0, 0, 0, 0, 1]]]
? nfisideal(bnf[7],[5,1;0,1])
% = 1
? nfisincl(x^2+1,x^4+1)
% = [x^2, -x^2]
? nfisincl(nfinit(x^2+1),nfinit(x^4+1),1)
% = [x^2, -x^2]
? nfisisom(x^3+x^2-2*x-1,x^3+x^2-2*x-1)
% = [x, x^2 - 2, -x^2 - x + 1]
? nfisisom(nfinit(x^3-2),nfinit(x^3-6*x^2-6*x-30),1)
% = [-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])
% = [[10951073973332888246310, 5442457637639729109215, 2693780223637146570
055, 3910837124677073032737, 3754666252923836621170; 0, 5, 0, 0, 0; 0, 0, 
5, 2, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 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]]
? 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 + 12991661155
2/29063006931199*x^10 + 1272919322296/29063006931199*x^9 - 2813750209005/2
9063006931199*x^8 - 2859411937992/29063006931199*x^7 - 414533880536/290630
06931199*x^6 - 35713977492936/29063006931199*x^5 - 17432607267590/29063006
931199*x^4 + 49785595543672/29063006931199*x^3 + 9423768373204/29063006931
199*x^2 - 42779776146743/29063006931199*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), Mod(-279140305176/29063006931199*x^11 + 129916611552/29
063006931199*x^10 + 1272919322296/29063006931199*x^9 - 2813750209005/29063
006931199*x^8 - 2859411937992/29063006931199*x^7 - 414533880536/2906300693
1199*x^6 - 35713977492936/29063006931199*x^5 - 17432607267590/290630069311
99*x^4 + 49785595543672/29063006931199*x^3 + 9423768373204/29063006931199*
x^2 - 13716769215544/29063006931199*x + 37962587857138/29063006931199, x^1
2 - 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]
? polred(x^5-2*x^4-4*x^3-96*x^2-352*x-568)
% = [x - 1, 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, x^5 - x^4 - 6*x^3 + 6*x^2 + 13*x - 
5]~
? polred(x^4-28*x^3-458*x^2+9156*x-25321,3)
% = 
[1 x - 1]

[1/4485*x^3 - 7/1495*x^2 - 1034/4485*x + 7924/4485 x^4 - 8*x^2 + 6]

[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]

? polred(x^4+576,1)
% = [x - 1, x^2 + 1, x^4 - x^2 + 1, x^2 - x + 1]~
? polred(x^4+576,3)
% = 
[1 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]

[1/192*x^3 + 1/8*x + 1/2 x^2 - x + 1]

? polred(p2,0,fa)
% = [x - 1, 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, x^5 - 2*x^4 - 53*x^3 - 46*x^2 + 508*x + 913, x^5 -
 2*x^4 - 62*x^3 + 85*x^2 + 818*x + 1]~
? polred(p2,1,fa)
% = [x - 1, 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, x^5 - 2*x^4 - 53*x^3 - 46*x^2 + 508*x + 913, x^5 -
 2*x^4 - 62*x^3 + 85*x^2 + 818*x + 1]~
? 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 + 33*x - 1, x^3 - 363*x - 2663]~
? polsubcyclo(31,5)
% = x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5
? poltschirnhaus(x^5-x-1)
% = x^5 - 10*x^4 + 38*x^3 - 119*x^2 + 45*x - 31
? ? ? aa=rnfpseudobasis(nf2,p)
% = [[[1, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [-2, 0, 0]~, [11, 0, 0]~; [0, 0,
 0]~, [1, 0, 0]~, [0, 0, 0]~, [2, 0, 0]~, [-8, 0, 0]~; [0, 0, 0]~, [0, 0, 
0]~, [1, 0, 0]~, [1, 0, 0]~, [4, 0, 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, 3/5; 0, 1, 2/5; 0, 0, 1/5], [1, 0, 8/2
5; 0, 1, 22/25; 0, 0, 1/25]], [416134375, 212940625, 388649575; 0, 3125, 5
50; 0, 0, 25], [-1280, 5, 5]~]
? rnfbasis(bnf2,aa)
% = 
[[1, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [4/5, -4/5, -2/5]~ [-187/25, -208/25, 61
/25]~]

[[0, 0, 0]~ [1, 0, 0]~ [0, 0, 0]~ [-4/5, 4/5, 2/5]~ [196/25, 214/25, -88/2
5]~]

[[0, 0, 0]~ [0, 0, 0]~ [1, 0, 0]~ [-2/5, 2/5, 1/5]~ [122/25, 123/25, -116/
25]~]

[[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [-2/5, 2/5, 1/5]~ [104/25, 111/25, -62/2
5]~]

[[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [3/25, 2/25, -9/25]~]

? ? ? ? ? rnfdisc(nf2,p)
% = [[416134375, 212940625, 388649575; 0, 3125, 550; 0, 0, 25], [-1280, 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 - 15*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]~ [-2/5, 2/5, -4/5]~ [11/25, 99/25, -33/25
]~]

[[0, 0, 0]~ [1, 0, 0]~ [0, 0, 0]~ [2/5, -2/5, 4/5]~ [-8/25, -72/25, 24/25]
~]

[[0, 0, 0]~ [0, 0, 0]~ [1, 0, 0]~ [1/5, -1/5, 2/5]~ [4/25, 36/25, -12/25]~
]

[[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [1/5, -1/5, 2/5]~ [-2/25, -18/25, 6/25]~
]

[[0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [0, 0, 0]~ [1/25, 9/25, -3/25]~]

? ? ? ? ? ? ? ? ? ? rnfisfree(bnf2,aa)
% = 1
? ? ? ? rnfsteinitz(nf2,aa)
% = [[[1, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [4/5, -4/5, -2/5]~, [39/125, 11/
125, 11/125]~; [0, 0, 0]~, [1, 0, 0]~, [0, 0, 0]~, [-4/5, 4/5, 2/5]~, [-42
/125, -8/125, -8/125]~; [0, 0, 0]~, [0, 0, 0]~, [1, 0, 0]~, [-2/5, 2/5, 1/
5]~, [-29/125, 4/125, 4/125]~; [0, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~, [-2/5, 
2/5, 1/5]~, [-23/125, -2/125, -2/125]~; [0, 0, 0]~, [0, 0, 0]~, [0, 0, 0]~
, [0, 0, 0]~, [-1/125, 1/125, 1/125]~], [[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, 108; 0, 125, 22; 0, 0, 1]], [416134375, 212940625, 38864
9575; 0, 3125, 550; 0, 0, 25], [-1280, 5, 5]~]
? ? nfz=zetakinit(x^2-2);
? zetak(nfz,-3)
% = 0.091666666666666666666666666666666666666
? zetak(nfz,1.5+3*I)
% = 0.88324345992059326405525724366416928890 - 0.2067536250233895222724230
8991427938845*I
? ? setrand(1);quadclassunit(1-10^7,,[1,1])
% = [2416, [1208, 2], [Qfb(277, 55, 9028), Qfb(1700, 1249, 1700)], 1, 0.99
984980753776002339750644800000000000]
? setrand(1);quadclassunit(10^9-3,,[0.5,0.5])
% = [4, [4], [Qfb(3, 1, -83333333, 0.E-48)], 2800.625251907016076486370621
7370745513, 0.99903694589643832327024650000000000000]
? sizebyte(%)
% = 176
? getheap
% = [204, 136061]
? print("Total time spent: ",gettime);
Total time spent: 24925
? \q
Good bye!
