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Properties

Label 36.0.32258992457...8125.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{48}\cdot 5^{27}\cdot 13^{24}$
Root discriminant $79.99$
Ramified primes $3, 5, 13$
Class number $204100$ (GRH)
Class group $[10, 20410]$ (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -21, 336, -4902, 68880, -471501, 2966770, -16999917, 79247853, -94040687, 180939411, -206283594, 279984346, -159459888, 279084165, -145690029, 288977244, -149538474, 179581154, -79515333, 86239137, -7098692, 20272767, -1125369, 4800131, -773109, 677004, -109299, 79572, -8742, 6766, -879, 546, -91, 27, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^35 + 27*x^34 - 91*x^33 + 546*x^32 - 879*x^31 + 6766*x^30 - 8742*x^29 + 79572*x^28 - 109299*x^27 + 677004*x^26 - 773109*x^25 + 4800131*x^24 - 1125369*x^23 + 20272767*x^22 - 7098692*x^21 + 86239137*x^20 - 79515333*x^19 + 179581154*x^18 - 149538474*x^17 + 288977244*x^16 - 145690029*x^15 + 279084165*x^14 - 159459888*x^13 + 279984346*x^12 - 206283594*x^11 + 180939411*x^10 - 94040687*x^9 + 79247853*x^8 - 16999917*x^7 + 2966770*x^6 - 471501*x^5 + 68880*x^4 - 4902*x^3 + 336*x^2 - 21*x + 1)
 
gp: K = bnfinit(x^36 - 3*x^35 + 27*x^34 - 91*x^33 + 546*x^32 - 879*x^31 + 6766*x^30 - 8742*x^29 + 79572*x^28 - 109299*x^27 + 677004*x^26 - 773109*x^25 + 4800131*x^24 - 1125369*x^23 + 20272767*x^22 - 7098692*x^21 + 86239137*x^20 - 79515333*x^19 + 179581154*x^18 - 149538474*x^17 + 288977244*x^16 - 145690029*x^15 + 279084165*x^14 - 159459888*x^13 + 279984346*x^12 - 206283594*x^11 + 180939411*x^10 - 94040687*x^9 + 79247853*x^8 - 16999917*x^7 + 2966770*x^6 - 471501*x^5 + 68880*x^4 - 4902*x^3 + 336*x^2 - 21*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 3 x^{35} + 27 x^{34} - 91 x^{33} + 546 x^{32} - 879 x^{31} + 6766 x^{30} - 8742 x^{29} + 79572 x^{28} - 109299 x^{27} + 677004 x^{26} - 773109 x^{25} + 4800131 x^{24} - 1125369 x^{23} + 20272767 x^{22} - 7098692 x^{21} + 86239137 x^{20} - 79515333 x^{19} + 179581154 x^{18} - 149538474 x^{17} + 288977244 x^{16} - 145690029 x^{15} + 279084165 x^{14} - 159459888 x^{13} + 279984346 x^{12} - 206283594 x^{11} + 180939411 x^{10} - 94040687 x^{9} + 79247853 x^{8} - 16999917 x^{7} + 2966770 x^{6} - 471501 x^{5} + 68880 x^{4} - 4902 x^{3} + 336 x^{2} - 21 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(322589924575014201866984217657349278106004681202180683612823486328125=3^{48}\cdot 5^{27}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.99$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(256,·)$, $\chi_{585}(1,·)$, $\chi_{585}(133,·)$, $\chi_{585}(391,·)$, $\chi_{585}(523,·)$, $\chi_{585}(16,·)$, $\chi_{585}(529,·)$, $\chi_{585}(274,·)$, $\chi_{585}(22,·)$, $\chi_{585}(412,·)$, $\chi_{585}(157,·)$, $\chi_{585}(289,·)$, $\chi_{585}(547,·)$, $\chi_{585}(406,·)$, $\chi_{585}(172,·)$, $\chi_{585}(562,·)$, $\chi_{585}(178,·)$, $\chi_{585}(568,·)$, $\chi_{585}(313,·)$, $\chi_{585}(61,·)$, $\chi_{585}(451,·)$, $\chi_{585}(196,·)$, $\chi_{585}(328,·)$, $\chi_{585}(334,·)$, $\chi_{585}(79,·)$, $\chi_{585}(211,·)$, $\chi_{585}(139,·)$, $\chi_{585}(469,·)$, $\chi_{585}(217,·)$, $\chi_{585}(94,·)$, $\chi_{585}(352,·)$, $\chi_{585}(484,·)$, $\chi_{585}(367,·)$, $\chi_{585}(373,·)$, $\chi_{585}(118,·)$, $\chi_{585}(508,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{19} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{23} - \frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{10} - \frac{1}{4} a^{5} + \frac{1}{4}$, $\frac{1}{4} a^{26} - \frac{1}{4} a^{11} - \frac{1}{4} a^{6} + \frac{1}{4} a$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{12} - \frac{1}{4} a^{7} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{28} - \frac{1}{4} a^{13} - \frac{1}{4} a^{8} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{29} - \frac{1}{4} a^{14} - \frac{1}{4} a^{9} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{30} - \frac{1}{4} a^{15} - \frac{1}{4} a^{10} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{31} - \frac{1}{4} a^{16} - \frac{1}{4} a^{11} + \frac{1}{4} a^{6}$, $\frac{1}{3452} a^{32} - \frac{94}{863} a^{31} + \frac{283}{3452} a^{30} + \frac{97}{1726} a^{29} - \frac{31}{3452} a^{28} - \frac{137}{3452} a^{27} - \frac{73}{1726} a^{26} - \frac{191}{1726} a^{25} + \frac{141}{3452} a^{24} - \frac{3}{863} a^{23} - \frac{55}{863} a^{22} + \frac{217}{3452} a^{21} + \frac{23}{1726} a^{20} - \frac{97}{863} a^{19} + \frac{45}{863} a^{18} + \frac{87}{3452} a^{17} - \frac{9}{3452} a^{16} - \frac{633}{1726} a^{15} - \frac{605}{1726} a^{14} - \frac{773}{1726} a^{13} + \frac{503}{3452} a^{12} + \frac{75}{3452} a^{11} - \frac{85}{863} a^{10} - \frac{847}{3452} a^{9} + \frac{991}{3452} a^{8} + \frac{1701}{3452} a^{7} + \frac{1013}{3452} a^{6} + \frac{1299}{3452} a^{5} + \frac{69}{3452} a^{4} - \frac{1699}{3452} a^{3} + \frac{49}{863} a^{2} - \frac{263}{3452} a + \frac{511}{3452}$, $\frac{1}{4056194047592548431350334740797081361870478221925374972} a^{33} - \frac{111479645312704389429722526635213222077597338669999}{4056194047592548431350334740797081361870478221925374972} a^{32} + \frac{145214282686042170207768997601220550157482044459420829}{2028097023796274215675167370398540680935239110962687486} a^{31} + \frac{140756182166255407581234694112513201525330536628758831}{4056194047592548431350334740797081361870478221925374972} a^{30} - \frac{269702033685692627586846385861900883016150069655325401}{4056194047592548431350334740797081361870478221925374972} a^{29} + \frac{40446162252585662531136991953064450587827088979251509}{4056194047592548431350334740797081361870478221925374972} a^{28} + \frac{41689048788033201780752429958706840904838173491542106}{1014048511898137107837583685199270340467619555481343743} a^{27} - \frac{63965615042030370173831112049093752440102154514777157}{2028097023796274215675167370398540680935239110962687486} a^{26} - \frac{175124986436512043566694003626056636337069759587073625}{4056194047592548431350334740797081361870478221925374972} a^{25} + \frac{126346216942321058715954825184117725726062028910577034}{1014048511898137107837583685199270340467619555481343743} a^{24} + \frac{268815075354719694488724096001857345133014780168874749}{4056194047592548431350334740797081361870478221925374972} a^{23} + \frac{76255657571278599071550789837297055651383334931099511}{4056194047592548431350334740797081361870478221925374972} a^{22} + \frac{334300817654255571439811543923841175605095414934881993}{4056194047592548431350334740797081361870478221925374972} a^{21} + \frac{114237112725879604964650774991332338449074216164279721}{1014048511898137107837583685199270340467619555481343743} a^{20} + \frac{603536686946641691413228283531074576014919774974014371}{4056194047592548431350334740797081361870478221925374972} a^{19} - \frac{221081788209552824953419481486046522540575919935075299}{4056194047592548431350334740797081361870478221925374972} a^{18} - \frac{479823246783743754102605935181546338369638823342255505}{4056194047592548431350334740797081361870478221925374972} a^{17} - \frac{574263201449938709110592737124974303264268208205299387}{4056194047592548431350334740797081361870478221925374972} a^{16} - \frac{275507892746955481790175160916021045415791182223378841}{1014048511898137107837583685199270340467619555481343743} a^{15} + \frac{434449395176242957542608225824120561878693474328472197}{2028097023796274215675167370398540680935239110962687486} a^{14} + \frac{315353248395108329662982940244347495198478017748763112}{1014048511898137107837583685199270340467619555481343743} a^{13} - \frac{122604089891523738567953582769364947172597455449065751}{2028097023796274215675167370398540680935239110962687486} a^{12} - \frac{668037886936897315225871860433999043417748033134561623}{2028097023796274215675167370398540680935239110962687486} a^{11} - \frac{946329546725325692656713883420200092596298965659045107}{2028097023796274215675167370398540680935239110962687486} a^{10} + \frac{122686392059654292174739931341492746749506427891499475}{1014048511898137107837583685199270340467619555481343743} a^{9} + \frac{492509963906012278818145338064277396090166366369962397}{2028097023796274215675167370398540680935239110962687486} a^{8} + \frac{1479412755189815797716205784777839983098818215675875637}{4056194047592548431350334740797081361870478221925374972} a^{7} + \frac{388737810130668857927511255088683462771199305140227219}{4056194047592548431350334740797081361870478221925374972} a^{6} - \frac{597108824327036013314211709800794205848095314820655985}{4056194047592548431350334740797081361870478221925374972} a^{5} - \frac{336947245160994045398016047987889712045674418274168242}{1014048511898137107837583685199270340467619555481343743} a^{4} + \frac{28235510952368963187185437831036766830659570896869747}{2028097023796274215675167370398540680935239110962687486} a^{3} + \frac{829182731726189900639424246944650891653686804632748171}{2028097023796274215675167370398540680935239110962687486} a^{2} + \frac{282354162391107073348148047613190300116507199368295381}{4056194047592548431350334740797081361870478221925374972} a + \frac{1916047629377078054886816772258056117244504078568133749}{4056194047592548431350334740797081361870478221925374972}$, $\frac{1}{4056194047592548431350334740797081361870478221925374972} a^{34} - \frac{176328832116744714022190587688137906440727537241869}{2028097023796274215675167370398540680935239110962687486} a^{32} + \frac{116519966738862015586186631386468374820696769126880513}{2028097023796274215675167370398540680935239110962687486} a^{31} + \frac{77145218377190551153916258930921781727893187344838325}{1014048511898137107837583685199270340467619555481343743} a^{30} + \frac{86482988520781305947813196348255473089839390561034058}{1014048511898137107837583685199270340467619555481343743} a^{29} - \frac{1972684736937546488206396007572591115859637829524233}{1014048511898137107837583685199270340467619555481343743} a^{28} - \frac{446448895284731318854947572140965240015997499395472319}{4056194047592548431350334740797081361870478221925374972} a^{27} + \frac{76620707073332695462573742860066613605063911995115990}{1014048511898137107837583685199270340467619555481343743} a^{26} + \frac{146805585261655276018731563546458447042711660218953325}{4056194047592548431350334740797081361870478221925374972} a^{25} + \frac{171365778243396660430935460742377044319365815238515721}{2028097023796274215675167370398540680935239110962687486} a^{24} - \frac{199471026330453124262111479569756484213058286959078617}{2028097023796274215675167370398540680935239110962687486} a^{23} + \frac{212563856388910269939943303918341301904653801493759951}{4056194047592548431350334740797081361870478221925374972} a^{22} - \frac{58068199978742879778880655882589098101425191855875241}{1014048511898137107837583685199270340467619555481343743} a^{21} - \frac{72989921178046724465092157434898179431343472369814191}{2028097023796274215675167370398540680935239110962687486} a^{20} + \frac{802363598780308750755970557024125920386027819539521177}{4056194047592548431350334740797081361870478221925374972} a^{19} + \frac{399456519887362695319108477944995030559878872550245419}{2028097023796274215675167370398540680935239110962687486} a^{18} + \frac{290130317671740418671587582208926475305257931422617699}{4056194047592548431350334740797081361870478221925374972} a^{17} + \frac{199652526162072103616832012994417744083505935232960857}{4056194047592548431350334740797081361870478221925374972} a^{16} - \frac{152281848744156663341864378512618003241996804861345669}{2028097023796274215675167370398540680935239110962687486} a^{15} - \frac{57880561257374508511149087661936114926513947400961187}{1014048511898137107837583685199270340467619555481343743} a^{14} + \frac{348020904813525843499712256471217368139900234891748633}{4056194047592548431350334740797081361870478221925374972} a^{13} - \frac{71817157713222345382139314080232945552075694125596055}{1014048511898137107837583685199270340467619555481343743} a^{12} - \frac{1972008360584807778824633533096448431031081059275485723}{4056194047592548431350334740797081361870478221925374972} a^{11} - \frac{1116871589827869211547668491510402984015070014870869579}{4056194047592548431350334740797081361870478221925374972} a^{10} + \frac{801082394708897419126774636815964920100532948498516553}{4056194047592548431350334740797081361870478221925374972} a^{9} - \frac{917309572807131422952059650349362260259931920255983021}{4056194047592548431350334740797081361870478221925374972} a^{8} - \frac{276662632812692099417147155170890455629960039234313843}{1014048511898137107837583685199270340467619555481343743} a^{7} + \frac{270172028361392782706335283508885911595989881914083496}{1014048511898137107837583685199270340467619555481343743} a^{6} - \frac{1695537735070626947073568344277510977351296738227884525}{4056194047592548431350334740797081361870478221925374972} a^{5} - \frac{848720430049582594886884933412828787185743445431876641}{2028097023796274215675167370398540680935239110962687486} a^{4} + \frac{627584290438293013890630157022188414000073531258415959}{4056194047592548431350334740797081361870478221925374972} a^{3} + \frac{216555893175631645739639398707108347822764727912301833}{2028097023796274215675167370398540680935239110962687486} a^{2} - \frac{543214307307301241857517730602870840911261570748490257}{4056194047592548431350334740797081361870478221925374972} a + \frac{741046463763159089899016661038948716336440637205793537}{2028097023796274215675167370398540680935239110962687486}$, $\frac{1}{4056194047592548431350334740797081361870478221925374972} a^{35} - \frac{545994934971933847298391110888687935739480726465943}{4056194047592548431350334740797081361870478221925374972} a^{32} - \frac{23795945328666489840498790620345601269924460258454822}{1014048511898137107837583685199270340467619555481343743} a^{31} + \frac{410341680045567203716847437413739769114417681560884255}{4056194047592548431350334740797081361870478221925374972} a^{30} + \frac{477643389548149759211865579709372963608411765795570375}{4056194047592548431350334740797081361870478221925374972} a^{29} + \frac{100069846800541874732107382734756083013667212975399702}{1014048511898137107837583685199270340467619555481343743} a^{28} - \frac{5278098868843540098878764435167485830328503167666123}{2028097023796274215675167370398540680935239110962687486} a^{27} + \frac{256432355545992498918500139706375901871326097681572967}{4056194047592548431350334740797081361870478221925374972} a^{26} - \frac{121863940185126080870210587861591186891599222840636941}{1014048511898137107837583685199270340467619555481343743} a^{25} - \frac{175401950395435833533275238143781613308721749541754805}{4056194047592548431350334740797081361870478221925374972} a^{24} - \frac{15337742880001194514773416919649393674433252441715577}{1014048511898137107837583685199270340467619555481343743} a^{23} - \frac{112955298782186296927425748625561979855088878531161627}{2028097023796274215675167370398540680935239110962687486} a^{22} + \frac{155625368907976555436155510498858398882889733213407715}{2028097023796274215675167370398540680935239110962687486} a^{21} - \frac{145507430291670045190959528776123952397897679198126439}{4056194047592548431350334740797081361870478221925374972} a^{20} - \frac{682864657252765625104424001191433349263238556231951575}{4056194047592548431350334740797081361870478221925374972} a^{19} + \frac{86766538912155164735480264669564524132237971722019725}{4056194047592548431350334740797081361870478221925374972} a^{18} - \frac{896495092120253264212280469069656137072464683809817677}{4056194047592548431350334740797081361870478221925374972} a^{17} - \frac{18147188117768867804471188254362881443354324106835965}{4056194047592548431350334740797081361870478221925374972} a^{16} + \frac{1635930725681114979333590632986306626928848569663399073}{4056194047592548431350334740797081361870478221925374972} a^{15} + \frac{467551095441717351853576804279483569162335068743374631}{2028097023796274215675167370398540680935239110962687486} a^{14} + \frac{1842850853434887338145842165091435904923728699572201217}{4056194047592548431350334740797081361870478221925374972} a^{13} + \frac{828488811463070751346989313179792821358794136351717213}{4056194047592548431350334740797081361870478221925374972} a^{12} - \frac{1655537068075610889369541179259284549534790372467558101}{4056194047592548431350334740797081361870478221925374972} a^{11} - \frac{1881171339985954786483529757956749287068200007897147287}{4056194047592548431350334740797081361870478221925374972} a^{10} - \frac{150552385691743440726535268322409894811096832294158930}{1014048511898137107837583685199270340467619555481343743} a^{9} + \frac{170496378341276994863796264199121910160350364210464534}{1014048511898137107837583685199270340467619555481343743} a^{8} + \frac{1996764264170779677992502275208740208051347167384389527}{4056194047592548431350334740797081361870478221925374972} a^{7} - \frac{247637880081470480747845982067997217351056304183502643}{4056194047592548431350334740797081361870478221925374972} a^{6} - \frac{937325303000257688613558382273959655094930238295775831}{2028097023796274215675167370398540680935239110962687486} a^{5} + \frac{234777721069374475229030039752735284965046220848772447}{2028097023796274215675167370398540680935239110962687486} a^{4} - \frac{774482962933129680532249610172774818664946521127732475}{4056194047592548431350334740797081361870478221925374972} a^{3} - \frac{722250377759408689196697496129863127994557417989601517}{4056194047592548431350334740797081361870478221925374972} a^{2} - \frac{1411965055418013011278832870238952898732882018450253657}{4056194047592548431350334740797081361870478221925374972} a - \frac{849367443160196545367180230890120769036341099198187357}{2028097023796274215675167370398540680935239110962687486}$

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

Class group and class number

$C_{10}\times C_{20410}$, which has order $204100$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{38764034711801584300232134112220550339577166999354}{1175027244377910901318173447507845122210451396849761} a^{35} + \frac{108538967364939431629652165406425149855542160824156}{1175027244377910901318173447507845122210451396849761} a^{34} - \frac{1023738045501640454775641992433827901866827702065666}{1175027244377910901318173447507845122210451396849761} a^{33} + \frac{3319299221915711366989909284195413687379110476949871}{1175027244377910901318173447507845122210451396849761} a^{32} - \frac{20469581525298455628651067383207148825116832055845836}{1175027244377910901318173447507845122210451396849761} a^{31} + \frac{29873941822126422976549220248038806089490616037877552}{1175027244377910901318173447507845122210451396849761} a^{30} - \frac{255663780843686023936517847914865210858969781521925266}{1175027244377910901318173447507845122210451396849761} a^{29} + \frac{286742066420485573526235312413658978947403878885780166}{1175027244377910901318173447507845122210451396849761} a^{28} - \frac{3019248454037087472952823826450620474183217775671927752}{1175027244377910901318173447507845122210451396849761} a^{27} + \frac{3623167253732476089481034744074650134036529029220531186}{1175027244377910901318173447507845122210451396849761} a^{26} - \frac{25425332869674661026163149852659019143464287924953613354}{1175027244377910901318173447507845122210451396849761} a^{25} + \frac{24760291701551796698066055747614666032026633883149262728}{1175027244377910901318173447507845122210451396849761} a^{24} - \frac{180328071413912958182137353691049517980765015746170920460}{1175027244377910901318173447507845122210451396849761} a^{23} + \frac{6693467074157249980880413791991344329819965202434249983}{1175027244377910901318173447507845122210451396849761} a^{22} - \frac{778899171027500984480776327017331990949045577832882324786}{1175027244377910901318173447507845122210451396849761} a^{21} + \frac{118416647591625807027838763916649092123845097184138669524}{1175027244377910901318173447507845122210451396849761} a^{20} - \frac{3295413463454535555913656371172786338359975445116169916120}{1175027244377910901318173447507845122210451396849761} a^{19} + \frac{2416348114883032150832588846585122674949697486642220319576}{1175027244377910901318173447507845122210451396849761} a^{18} - \frac{6376594377344349823755303142844478708329924357071198209000}{1175027244377910901318173447507845122210451396849761} a^{17} + \frac{4433789455399610674289506828958331889201949987162216071362}{1175027244377910901318173447507845122210451396849761} a^{16} - \frac{10108839386572626117234449116763942852637005614550407510744}{1175027244377910901318173447507845122210451396849761} a^{15} + \frac{3462336704326244582389328355411851827467219848595649804014}{1175027244377910901318173447507845122210451396849761} a^{14} - \frac{9795567826481469910834321211137683080665111478100738170256}{1175027244377910901318173447507845122210451396849761} a^{13} + \frac{4071488697531826612594202565588345823321608817260725414626}{1175027244377910901318173447507845122210451396849761} a^{12} - \frac{9720053406806432428803652747350709891481141758777415215590}{1175027244377910901318173447507845122210451396849761} a^{11} + \frac{5884563450781320267471359440996097932038379795637548841520}{1175027244377910901318173447507845122210451396849761} a^{10} - \frac{5518007134208799941241633639662149337821233755346570480254}{1175027244377910901318173447507845122210451396849761} a^{9} + \frac{2318753582811125102628097406074635048468244062490531478196}{1175027244377910901318173447507845122210451396849761} a^{8} - \frac{2409809462817878724592064667318430751826641120480941681344}{1175027244377910901318173447507845122210451396849761} a^{7} + \frac{79309759782914087725524048985950214700416249571804742094}{1175027244377910901318173447507845122210451396849761} a^{6} - \frac{12463666532604322327872016043360522837182833489431575260}{1175027244377910901318173447507845122210451396849761} a^{5} + \frac{1552529714081921664309232158550488990510921052913423664}{1175027244377910901318173447507845122210451396849761} a^{4} - \frac{109885905801333543227978802973644699326730023397165386}{1175027244377910901318173447507845122210451396849761} a^{3} - \frac{214772700641592902275859248177999215927631487175951111}{1175027244377910901318173447507845122210451396849761} a^{2} - \frac{449660591828907407721846048345565821845031425449716}{1175027244377910901318173447507845122210451396849761} a + \frac{18828149562032652959320600957546252640073566202990}{1175027244377910901318173447507845122210451396849761} \) (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:  \( 478341710831896.44 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.169.1, 3.3.13689.1, \(\Q(\zeta_{5})\), 6.6.820125.1, 6.6.23423590125.2, 6.6.3570125.1, 6.6.23423590125.1, 9.9.2565164201769.1, 12.0.84075626953125.1, 12.0.68583071792999689453125.2, 12.0.1593224064453125.1, 12.0.68583071792999689453125.1, 18.18.12851694105541388560018283203125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
13Data not computed