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Properties

Label 32.0.91066857695...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $64.75$
Ramified primes $2, 5, 13$
Class number $480$ (GRH)
Class group $[4, 120]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43046721, -19131876, 8503056, -36728478, 9710280, 1679616, 18148455, 3806838, -5568993, -7283844, 13451220, -5443884, 7294861, -10470992, 3473250, -2313570, 4068117, 615296, -386025, -575216, 191742, -80556, -32711, 19320, 11736, -3920, -741, 332, 21, 36, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 9*x^30 + 36*x^29 + 21*x^28 + 332*x^27 - 741*x^26 - 3920*x^25 + 11736*x^24 + 19320*x^23 - 32711*x^22 - 80556*x^21 + 191742*x^20 - 575216*x^19 - 386025*x^18 + 615296*x^17 + 4068117*x^16 - 2313570*x^15 + 3473250*x^14 - 10470992*x^13 + 7294861*x^12 - 5443884*x^11 + 13451220*x^10 - 7283844*x^9 - 5568993*x^8 + 3806838*x^7 + 18148455*x^6 + 1679616*x^5 + 9710280*x^4 - 36728478*x^3 + 8503056*x^2 - 19131876*x + 43046721)
 
gp: K = bnfinit(x^32 - 2*x^31 - 9*x^30 + 36*x^29 + 21*x^28 + 332*x^27 - 741*x^26 - 3920*x^25 + 11736*x^24 + 19320*x^23 - 32711*x^22 - 80556*x^21 + 191742*x^20 - 575216*x^19 - 386025*x^18 + 615296*x^17 + 4068117*x^16 - 2313570*x^15 + 3473250*x^14 - 10470992*x^13 + 7294861*x^12 - 5443884*x^11 + 13451220*x^10 - 7283844*x^9 - 5568993*x^8 + 3806838*x^7 + 18148455*x^6 + 1679616*x^5 + 9710280*x^4 - 36728478*x^3 + 8503056*x^2 - 19131876*x + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} - 9 x^{30} + 36 x^{29} + 21 x^{28} + 332 x^{27} - 741 x^{26} - 3920 x^{25} + 11736 x^{24} + 19320 x^{23} - 32711 x^{22} - 80556 x^{21} + 191742 x^{20} - 575216 x^{19} - 386025 x^{18} + 615296 x^{17} + 4068117 x^{16} - 2313570 x^{15} + 3473250 x^{14} - 10470992 x^{13} + 7294861 x^{12} - 5443884 x^{11} + 13451220 x^{10} - 7283844 x^{9} - 5568993 x^{8} + 3806838 x^{7} + 18148455 x^{6} + 1679616 x^{5} + 9710280 x^{4} - 36728478 x^{3} + 8503056 x^{2} - 19131876 x + 43046721 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9106685769537214956799814036094976000000000000000000000000=2^{48}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.75$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(520=2^{3}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(259,·)$, $\chi_{520}(51,·)$, $\chi_{520}(129,·)$, $\chi_{520}(131,·)$, $\chi_{520}(281,·)$, $\chi_{520}(27,·)$, $\chi_{520}(161,·)$, $\chi_{520}(291,·)$, $\chi_{520}(177,·)$, $\chi_{520}(307,·)$, $\chi_{520}(57,·)$, $\chi_{520}(187,·)$, $\chi_{520}(411,·)$, $\chi_{520}(313,·)$, $\chi_{520}(83,·)$, $\chi_{520}(417,·)$, $\chi_{520}(73,·)$, $\chi_{520}(339,·)$, $\chi_{520}(203,·)$, $\chi_{520}(337,·)$, $\chi_{520}(467,·)$, $\chi_{520}(441,·)$, $\chi_{520}(473,·)$, $\chi_{520}(99,·)$, $\chi_{520}(209,·)$, $\chi_{520}(233,·)$, $\chi_{520}(363,·)$, $\chi_{520}(369,·)$, $\chi_{520}(499,·)$, $\chi_{520}(489,·)$, $\chi_{520}(443,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{16} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{2}{9} a^{8} - \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{19} - \frac{4}{27} a^{17} + \frac{1}{9} a^{15} + \frac{4}{27} a^{13} - \frac{1}{27} a^{11} - \frac{2}{27} a^{9} + \frac{1}{9} a^{7} - \frac{4}{27} a^{5} - \frac{5}{27} a^{3}$, $\frac{1}{54} a^{20} + \frac{1}{27} a^{18} + \frac{1}{9} a^{16} + \frac{2}{27} a^{14} - \frac{2}{27} a^{12} + \frac{1}{54} a^{10} - \frac{2}{27} a^{6} - \frac{1}{27} a^{4} + \frac{1}{9} a^{2} - \frac{1}{2}$, $\frac{1}{54} a^{21} - \frac{2}{27} a^{17} - \frac{1}{27} a^{15} + \frac{1}{9} a^{13} + \frac{1}{18} a^{11} + \frac{2}{27} a^{9} + \frac{4}{27} a^{7} + \frac{1}{9} a^{5} - \frac{1}{27} a^{3} - \frac{1}{2} a$, $\frac{1}{54} a^{22} + \frac{1}{27} a^{18} - \frac{4}{27} a^{16} + \frac{1}{9} a^{14} - \frac{1}{6} a^{12} - \frac{1}{27} a^{10} - \frac{2}{27} a^{8} + \frac{1}{9} a^{6} - \frac{4}{27} a^{4} - \frac{7}{18} a^{2}$, $\frac{1}{54} a^{23} + \frac{1}{54} a^{13} + \frac{25}{54} a^{3}$, $\frac{1}{162} a^{24} + \frac{1}{162} a^{23} + \frac{1}{81} a^{19} - \frac{13}{81} a^{17} - \frac{2}{27} a^{15} - \frac{17}{162} a^{14} - \frac{1}{18} a^{13} - \frac{1}{81} a^{11} - \frac{2}{81} a^{9} + \frac{13}{27} a^{7} + \frac{32}{81} a^{5} - \frac{11}{162} a^{4} - \frac{7}{54} a^{3} + \frac{1}{3} a$, $\frac{1}{486} a^{25} + \frac{1}{486} a^{24} + \frac{1}{162} a^{23} + \frac{1}{162} a^{21} - \frac{1}{486} a^{20} - \frac{1}{81} a^{19} - \frac{7}{243} a^{18} + \frac{11}{81} a^{17} + \frac{1}{81} a^{16} + \frac{13}{486} a^{15} - \frac{7}{162} a^{14} + \frac{17}{162} a^{13} + \frac{14}{243} a^{12} - \frac{13}{162} a^{11} - \frac{25}{486} a^{10} - \frac{5}{81} a^{9} - \frac{20}{81} a^{8} - \frac{17}{81} a^{7} - \frac{70}{243} a^{6} - \frac{77}{486} a^{5} - \frac{11}{162} a^{4} - \frac{1}{54} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{307638} a^{26} - \frac{251}{307638} a^{25} + \frac{71}{51273} a^{24} - \frac{7}{1899} a^{23} + \frac{259}{102546} a^{22} + \frac{886}{153819} a^{21} + \frac{53}{51273} a^{20} + \frac{2765}{153819} a^{19} + \frac{2816}{51273} a^{18} + \frac{7948}{51273} a^{17} + \frac{23413}{307638} a^{16} - \frac{2371}{102546} a^{15} + \frac{4282}{51273} a^{14} - \frac{11038}{153819} a^{13} + \frac{1895}{102546} a^{12} + \frac{12430}{153819} a^{11} + \frac{6883}{51273} a^{10} - \frac{3467}{51273} a^{9} + \frac{475}{51273} a^{8} + \frac{26354}{153819} a^{7} - \frac{113531}{307638} a^{6} - \frac{49103}{102546} a^{5} + \frac{2489}{5697} a^{4} + \frac{284}{633} a^{3} + \frac{145}{1266} a^{2} + \frac{50}{211} a - \frac{89}{211}$, $\frac{1}{922914} a^{27} + \frac{1}{922914} a^{26} - \frac{53}{307638} a^{25} - \frac{25}{11394} a^{24} - \frac{973}{153819} a^{23} + \frac{1939}{461457} a^{22} + \frac{416}{153819} a^{21} + \frac{14}{2187} a^{20} - \frac{2299}{153819} a^{19} - \frac{5939}{153819} a^{18} + \frac{134059}{922914} a^{17} + \frac{31139}{307638} a^{16} - \frac{21127}{307638} a^{15} + \frac{43183}{922914} a^{14} - \frac{482}{153819} a^{13} - \frac{34658}{461457} a^{12} - \frac{10538}{153819} a^{11} + \frac{4858}{153819} a^{10} + \frac{23119}{153819} a^{9} + \frac{106301}{461457} a^{8} - \frac{196277}{922914} a^{7} - \frac{56525}{307638} a^{6} - \frac{7057}{34182} a^{5} + \frac{14879}{34182} a^{4} - \frac{2725}{5697} a^{3} - \frac{781}{1899} a^{2} + \frac{91}{211} a - \frac{91}{211}$, $\frac{1}{1226552706} a^{28} + \frac{103}{1226552706} a^{27} - \frac{73}{408850902} a^{26} + \frac{7259}{68141817} a^{25} - \frac{1162937}{408850902} a^{24} - \frac{2665049}{613276353} a^{23} + \frac{442325}{136283634} a^{22} - \frac{3414391}{613276353} a^{21} - \frac{746009}{204425451} a^{20} + \frac{231226}{204425451} a^{19} - \frac{26963447}{1226552706} a^{18} - \frac{7546931}{45427878} a^{17} - \frac{5321653}{408850902} a^{16} - \frac{21158482}{613276353} a^{15} + \frac{15626915}{136283634} a^{14} + \frac{67867813}{613276353} a^{13} - \frac{14225033}{408850902} a^{12} - \frac{28682351}{204425451} a^{11} - \frac{10911779}{204425451} a^{10} + \frac{67675808}{613276353} a^{9} + \frac{241877329}{1226552706} a^{8} + \frac{92810309}{408850902} a^{7} + \frac{67502377}{136283634} a^{6} - \frac{1757104}{7571313} a^{5} + \frac{4819819}{15142626} a^{4} + \frac{216989}{841257} a^{3} - \frac{72847}{186946} a^{2} + \frac{64727}{280419} a + \frac{39231}{93473}$, $\frac{1}{910391577421676936141664056557560726} a^{29} - \frac{378021417283641049181569}{455195788710838468070832028278780363} a^{28} + \frac{16788310074930495722308408343}{101154619713519659571296006284173414} a^{27} - \frac{16787764043994419351904034966}{50577309856759829785648003142086707} a^{26} - \frac{6371878792635619616893998959651}{303463859140558978713888018852520242} a^{25} - \frac{2214986726300139824641010401865041}{910391577421676936141664056557560726} a^{24} - \frac{651673932235708082828911782249953}{151731929570279489356944009426260121} a^{23} + \frac{1677020554323957672729167341250207}{455195788710838468070832028278780363} a^{22} - \frac{534519356427667464098293815345931}{101154619713519659571296006284173414} a^{21} + \frac{302526005462490187679708917569328}{151731929570279489356944009426260121} a^{20} - \frac{5004648902487414950094859962107063}{910391577421676936141664056557560726} a^{19} + \frac{2553930579312953143390300688433443}{151731929570279489356944009426260121} a^{18} + \frac{24047627663657314355135120208275729}{303463859140558978713888018852520242} a^{17} + \frac{45321982125065335269622645793291201}{455195788710838468070832028278780363} a^{16} - \frac{35464409159764532311534062464086421}{303463859140558978713888018852520242} a^{15} - \frac{88478064376033839707641169038368931}{910391577421676936141664056557560726} a^{14} - \frac{799264733645402569427981256424427}{5619701095195536642849778126898523} a^{13} - \frac{24207784967238086519264461499891504}{151731929570279489356944009426260121} a^{12} - \frac{36730504799570583841953290348686825}{303463859140558978713888018852520242} a^{11} + \frac{51735802321095673686725739774618065}{455195788710838468070832028278780363} a^{10} + \frac{112495350300206873824885871692069939}{910391577421676936141664056557560726} a^{9} - \frac{8917890359601394485335483848796792}{50577309856759829785648003142086707} a^{8} - \frac{26365247238205268490875496124157171}{101154619713519659571296006284173414} a^{7} - \frac{326380394295899435183084128590544}{5619701095195536642849778126898523} a^{6} - \frac{1827230291298467457505418860129235}{11239402190391073285699556253797046} a^{5} - \frac{86943718403181642168996809263729}{1248822465599008142855506250421894} a^{4} + \frac{150861563145362899366308688226624}{624411232799504071427753125210947} a^{3} - \frac{24033224285256745413795410468527}{69379025866611563491972569467883} a^{2} + \frac{4540898818669026458203886567557}{138758051733223126983945138935766} a - \frac{3389458906884773809885351981007}{7708780651845729276885841051987}$, $\frac{1}{8193524196795092425274976509018046534} a^{30} - \frac{1}{4096762098397546212637488254509023267} a^{29} + \frac{6599951310369366027041188}{455195788710838468070832028278780363} a^{28} - \frac{47351553670605865884468852407}{910391577421676936141664056557560726} a^{27} + \frac{141797262910713192378351950885}{1365587366132515404212496084836341089} a^{26} + \frac{2621858776239722061774930895158911}{8193524196795092425274976509018046534} a^{25} + \frac{566670213362794011050613522376463}{2731174732265030808424992169672682178} a^{24} - \frac{1836728653902261181075292743523842}{4096762098397546212637488254509023267} a^{23} + \frac{959689366115541140056621581673015}{455195788710838468070832028278780363} a^{22} + \frac{15470289454237208619912450341947013}{2731174732265030808424992169672682178} a^{21} + \frac{24247532773999099153398225639817352}{4096762098397546212637488254509023267} a^{20} + \frac{18821167801616175107582135895190427}{1365587366132515404212496084836341089} a^{19} - \frac{40075337693262973572088765632802901}{1365587366132515404212496084836341089} a^{18} - \frac{406407808625081650444624468023437165}{8193524196795092425274976509018046534} a^{17} - \frac{70311598365771772777120934319765877}{1365587366132515404212496084836341089} a^{16} + \frac{105412299807196345537486262728862219}{8193524196795092425274976509018046534} a^{15} - \frac{13696536046209371867842194494831335}{303463859140558978713888018852520242} a^{14} + \frac{204102027227474568089889905596147624}{1365587366132515404212496084836341089} a^{13} - \frac{161586996490756392151209592795712834}{1365587366132515404212496084836341089} a^{12} - \frac{346821389423972185210215791910564317}{8193524196795092425274976509018046534} a^{11} - \frac{578944269491690580819984005083200607}{4096762098397546212637488254509023267} a^{10} + \frac{14304863446825422146516175093209953}{455195788710838468070832028278780363} a^{9} - \frac{127003717864878642966582829616339005}{455195788710838468070832028278780363} a^{8} + \frac{18281730076854224040681711454912465}{101154619713519659571296006284173414} a^{7} + \frac{697483693028293242867409589537206}{1744045167474476889160275970416783} a^{6} - \frac{950421540631206056840154885867835}{11239402190391073285699556253797046} a^{5} - \frac{75547806829687892764498870231325}{11239402190391073285699556253797046} a^{4} - \frac{155936410983359303137568832165277}{624411232799504071427753125210947} a^{3} + \frac{30030180863602768621652195847137}{624411232799504071427753125210947} a^{2} + \frac{12002176355117652657216932682803}{138758051733223126983945138935766} a - \frac{3209920794405614539393651487437}{15417561303691458553771682103974}$, $\frac{1}{73741717771155831827474788581162418806} a^{31} - \frac{1}{36870858885577915913737394290581209403} a^{30} - \frac{1}{8193524196795092425274976509018046534} a^{29} - \frac{58223633791239217537272052}{4096762098397546212637488254509023267} a^{28} + \frac{725935566409723690124352448580}{12290286295192638637912464763527069801} a^{27} + \frac{26246364749998859330552165323468}{36870858885577915913737394290581209403} a^{26} - \frac{17072771895456437442492403849797907}{24580572590385277275824929527054139602} a^{25} + \frac{211743049558293676848572305013594479}{73741717771155831827474788581162418806} a^{24} - \frac{1462390896117182609363031085120817}{4096762098397546212637488254509023267} a^{23} - \frac{19141232891573927942831980166042587}{24580572590385277275824929527054139602} a^{22} - \frac{216390563024689057954187627429301754}{36870858885577915913737394290581209403} a^{21} - \frac{36378785899900378263331482902476043}{24580572590385277275824929527054139602} a^{20} - \frac{153359752424813529124892634178380535}{24580572590385277275824929527054139602} a^{19} - \frac{1505726101933598790553910362615603894}{36870858885577915913737394290581209403} a^{18} - \frac{64057978474730868710998001502930421}{12290286295192638637912464763527069801} a^{17} - \frac{1605141540129113803979254124415124514}{36870858885577915913737394290581209403} a^{16} + \frac{251812512979981137985954405651276169}{2731174732265030808424992169672682178} a^{15} - \frac{470025854878376604438081701513667403}{24580572590385277275824929527054139602} a^{14} - \frac{1651216954583031265356381652373730482}{12290286295192638637912464763527069801} a^{13} + \frac{1697609115092849753569211744910335959}{73741717771155831827474788581162418806} a^{12} - \frac{2795353372987424310227835708777134731}{36870858885577915913737394290581209403} a^{11} + \frac{348190792517682962440673412031171673}{8193524196795092425274976509018046534} a^{10} - \frac{1208443284979236470955274821847770953}{8193524196795092425274976509018046534} a^{9} + \frac{74452302356731483674254314501661618}{455195788710838468070832028278780363} a^{8} + \frac{140167068921848052001187206399011842}{455195788710838468070832028278780363} a^{7} + \frac{2567765699772545042502463960175215}{50577309856759829785648003142086707} a^{6} - \frac{9439100615383420386322286457269039}{101154619713519659571296006284173414} a^{5} + \frac{1462482936891677521064241859395919}{11239402190391073285699556253797046} a^{4} - \frac{1055193026457333774932287103257972}{5619701095195536642849778126898523} a^{3} + \frac{309960895774413151689101147032883}{1248822465599008142855506250421894} a^{2} + \frac{45183511761301913402195958702961}{138758051733223126983945138935766} a + \frac{7100392926066962857763422918821}{15417561303691458553771682103974}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}\times C_{120}$, which has order $480$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{89173462059473888207612393}{9247769973809359396472885450358969} a^{31} + \frac{140861654202766831672815232}{9247769973809359396472885450358969} a^{30} + \frac{2271962164560755349561536}{35432068865169959373459331227429} a^{29} - \frac{94286429829271347006803744}{342509999029976273943440201865147} a^{28} - \frac{308986854380262727540368896}{3082589991269786465490961816786323} a^{27} - \frac{76241306809845193930880438783}{18495539947618718792945770900717938} a^{26} + \frac{16005973449330521437661021120}{3082589991269786465490961816786323} a^{25} + \frac{260579292521048993684936029216}{9247769973809359396472885450358969} a^{24} - \frac{99966335240673235380707584}{1118095753090238108629293368439} a^{23} - \frac{377410402909256716025438894944}{3082589991269786465490961816786323} a^{22} - \frac{38676904652515126816309284509}{9247769973809359396472885450358969} a^{21} + \frac{1267882117706116887354912534016}{3082589991269786465490961816786323} a^{20} - \frac{4648180128928874640603751763968}{3082589991269786465490961816786323} a^{19} + \frac{46817187208040122830362116803232}{9247769973809359396472885450358969} a^{18} + \frac{8919675746860069726516249543744}{3082589991269786465490961816786323} a^{17} + \frac{346514758040782462264804052100481}{18495539947618718792945770900717938} a^{16} - \frac{8066890204467861084547627883072}{342509999029976273943440201865147} a^{15} + \frac{57767935787039064698991941369504}{3082589991269786465490961816786323} a^{14} - \frac{155868982138740343975065780848128}{3082589991269786465490961816786323} a^{13} + \frac{457769190070340359728343747576096}{9247769973809359396472885450358969} a^{12} - \frac{4400135665653539421057324030193961}{9247769973809359396472885450358969} a^{11} + \frac{54671395626340204765504264959040}{1027529997089928821830320605595441} a^{10} - \frac{387137808916823590054586611328}{12685555519628750886794081550561} a^{9} + \frac{795108374901587026286977727008}{114169999676658757981146733955049} a^{8} + \frac{470580163334646451777933144000}{12685555519628750886794081550561} a^{7} - \frac{9472711940209203875559709919533}{25371111039257501773588163101122} a^{6} - \frac{19318494285260102737321740608}{469835389615879662473854872243} a^{5} + \frac{36343442765396122949261110624}{1409506168847638987421564616729} a^{4} - \frac{30083051020948961583544298176}{156611796538626554157951624081} a^{3} + \frac{1596053420603930633066979040}{17401310726514061573105736009} a^{2} - \frac{284506567512047297176482276397}{156611796538626554157951624081} a + \frac{3680578706588423666289688320}{17401310726514061573105736009} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 68099381310471.18 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-130}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{65})\), 4.4.274625.1, 4.0.17576000.2, \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-10}, \sqrt{-26})\), 4.4.274625.2, 4.0.17576000.1, \(\Q(\sqrt{-10}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-26})\), 4.0.54925.1, 4.4.3515200.1, 4.0.2197.1, 4.4.140608.1, 4.0.21125.1, 4.4.1352000.1, \(\Q(\zeta_{5})\), 4.4.8000.1, 8.0.308915776000000.29, 8.0.73116160000.1, 8.0.308915776000000.41, 8.8.75418890625.1, 8.0.308915776000000.26, 8.0.308915776000000.53, 8.0.308915776000000.67, 8.0.12356631040000.1, 8.0.19770609664.1, 8.0.1827904000000.1, 8.0.64000000.1, 8.0.3016755625.1, 8.8.12356631040000.1, 8.0.446265625.1, 8.8.1827904000000.1, 8.0.12356631040000.3, 8.0.12356631040000.6, 8.0.1827904000000.4, 8.0.1827904000000.6, 16.0.95428956661682176000000000000.2, 16.0.152686330658691481600000000.1, 16.0.3341233033216000000000000.2, 16.0.5688009063105712890625.1, 16.16.95428956661682176000000000000.2, 16.0.95428956661682176000000000000.3, 16.0.95428956661682176000000000000.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
13Data not computed