Properties

Label 38.0.360...904.1
Degree $38$
Signature $[0, 19]$
Discriminant $-3.600\times 10^{93}$
Root discriminant $289.74$
Ramified primes $2, 191$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 + 181*x^36 + 14302*x^34 + 655785*x^32 + 19564842*x^30 + 403491764*x^28 + 5961216274*x^26 + 64460499272*x^24 + 516273473421*x^22 + 3076728313208*x^20 + 13620343868498*x^18 + 44444158242663*x^16 + 105381413708670*x^14 + 177560394917737*x^12 + 205585970435602*x^10 + 155336297697903*x^8 + 70268784621098*x^6 + 16089414412134*x^4 + 1207108975793*x^2 + 13841287201)
 
gp: K = bnfinit(x^38 + 181*x^36 + 14302*x^34 + 655785*x^32 + 19564842*x^30 + 403491764*x^28 + 5961216274*x^26 + 64460499272*x^24 + 516273473421*x^22 + 3076728313208*x^20 + 13620343868498*x^18 + 44444158242663*x^16 + 105381413708670*x^14 + 177560394917737*x^12 + 205585970435602*x^10 + 155336297697903*x^8 + 70268784621098*x^6 + 16089414412134*x^4 + 1207108975793*x^2 + 13841287201, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13841287201, 0, 1207108975793, 0, 16089414412134, 0, 70268784621098, 0, 155336297697903, 0, 205585970435602, 0, 177560394917737, 0, 105381413708670, 0, 44444158242663, 0, 13620343868498, 0, 3076728313208, 0, 516273473421, 0, 64460499272, 0, 5961216274, 0, 403491764, 0, 19564842, 0, 655785, 0, 14302, 0, 181, 0, 1]);
 

\( x^{38} + 181 x^{36} + 14302 x^{34} + 655785 x^{32} + 19564842 x^{30} + 403491764 x^{28} + 5961216274 x^{26} + 64460499272 x^{24} + 516273473421 x^{22} + 3076728313208 x^{20} + 13620343868498 x^{18} + 44444158242663 x^{16} + 105381413708670 x^{14} + 177560394917737 x^{12} + 205585970435602 x^{10} + 155336297697903 x^{8} + 70268784621098 x^{6} + 16089414412134 x^{4} + 1207108975793 x^{2} + 13841287201 \)

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

Invariants

Degree:  $38$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 19]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-36\!\cdots\!904\)\(\medspace = -\,2^{38}\cdot 191^{36}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $289.74$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 191$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $38$
This field is Galois and abelian over $\Q$.
Conductor:  \(764=2^{2}\cdot 191\)
Dirichlet character group:    $\lbrace$$\chi_{764}(1,·)$, $\chi_{764}(387,·)$, $\chi_{764}(5,·)$, $\chi_{764}(407,·)$, $\chi_{764}(579,·)$, $\chi_{764}(535,·)$, $\chi_{764}(25,·)$, $\chi_{764}(709,·)$, $\chi_{764}(197,·)$, $\chi_{764}(625,·)$, $\chi_{764}(559,·)$, $\chi_{764}(177,·)$, $\chi_{764}(243,·)$, $\chi_{764}(351,·)$, $\chi_{764}(451,·)$, $\chi_{764}(69,·)$, $\chi_{764}(327,·)$, $\chi_{764}(723,·)$, $\chi_{764}(341,·)$, $\chi_{764}(727,·)$, $\chi_{764}(345,·)$, $\chi_{764}(603,·)$, $\chi_{764}(733,·)$, $\chi_{764}(223,·)$, $\chi_{764}(609,·)$, $\chi_{764}(227,·)$, $\chi_{764}(489,·)$, $\chi_{764}(107,·)$, $\chi_{764}(753,·)$, $\chi_{764}(221,·)$, $\chi_{764}(371,·)$, $\chi_{764}(503,·)$, $\chi_{764}(153,·)$, $\chi_{764}(121,·)$, $\chi_{764}(507,·)$, $\chi_{764}(125,·)$, $\chi_{764}(605,·)$, $\chi_{764}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{1}{7} a$, $\frac{1}{49} a^{14} + \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} + \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{49} a^{16} + \frac{2}{49} a^{10} + \frac{1}{49} a^{4}$, $\frac{1}{49} a^{17} + \frac{2}{49} a^{11} + \frac{1}{49} a^{5}$, $\frac{1}{49} a^{18} + \frac{2}{49} a^{12} + \frac{1}{49} a^{6}$, $\frac{1}{49} a^{19} + \frac{2}{49} a^{13} + \frac{1}{49} a^{7}$, $\frac{1}{49} a^{20} - \frac{3}{49} a^{8} - \frac{2}{49} a^{2}$, $\frac{1}{343} a^{21} + \frac{3}{343} a^{15} + \frac{3}{343} a^{9} + \frac{1}{343} a^{3}$, $\frac{1}{343} a^{22} + \frac{3}{343} a^{16} + \frac{3}{343} a^{10} + \frac{1}{343} a^{4}$, $\frac{1}{343} a^{23} + \frac{3}{343} a^{17} + \frac{3}{343} a^{11} + \frac{1}{343} a^{5}$, $\frac{1}{343} a^{24} + \frac{3}{343} a^{18} + \frac{3}{343} a^{12} + \frac{1}{343} a^{6}$, $\frac{1}{343} a^{25} + \frac{3}{343} a^{19} + \frac{3}{343} a^{13} + \frac{1}{343} a^{7}$, $\frac{1}{2401} a^{26} - \frac{1}{2401} a^{24} + \frac{1}{2401} a^{22} - \frac{11}{2401} a^{20} + \frac{11}{2401} a^{18} - \frac{11}{2401} a^{16} + \frac{24}{2401} a^{14} - \frac{73}{2401} a^{12} + \frac{73}{2401} a^{10} - \frac{13}{2401} a^{8} + \frac{258}{2401} a^{6} - \frac{258}{2401} a^{4} + \frac{6}{49} a^{2}$, $\frac{1}{2401} a^{27} - \frac{1}{2401} a^{25} + \frac{1}{2401} a^{23} + \frac{3}{2401} a^{21} + \frac{11}{2401} a^{19} - \frac{11}{2401} a^{17} + \frac{17}{2401} a^{15} - \frac{73}{2401} a^{13} + \frac{73}{2401} a^{11} - \frac{69}{2401} a^{9} - \frac{85}{2401} a^{7} - \frac{258}{2401} a^{5} + \frac{37}{343} a^{3} - \frac{1}{7} a$, $\frac{1}{2401} a^{28} - \frac{3}{2401} a^{22} - \frac{15}{2401} a^{16} - \frac{17}{2401} a^{10} - \frac{6}{2401} a^{4}$, $\frac{1}{16807} a^{29} + \frac{1}{16807} a^{27} + \frac{13}{16807} a^{25} + \frac{19}{16807} a^{23} - \frac{11}{16807} a^{21} + \frac{4}{16807} a^{19} + \frac{135}{16807} a^{17} - \frac{74}{16807} a^{15} - \frac{815}{16807} a^{13} - \frac{53}{2401} a^{11} - \frac{895}{16807} a^{9} - \frac{463}{16807} a^{7} - \frac{3232}{16807} a^{5} + \frac{137}{343} a^{3} + \frac{3}{7} a$, $\frac{1}{16807} a^{30} + \frac{1}{16807} a^{28} - \frac{1}{16807} a^{26} - \frac{16}{16807} a^{24} + \frac{24}{16807} a^{22} + \frac{158}{16807} a^{20} - \frac{166}{16807} a^{18} - \frac{116}{16807} a^{16} - \frac{122}{16807} a^{14} + \frac{72}{2401} a^{12} - \frac{55}{16807} a^{10} - \frac{624}{16807} a^{8} - \frac{6893}{16807} a^{6} - \frac{625}{2401} a^{4} + \frac{5}{49} a^{2}$, $\frac{1}{16807} a^{31} - \frac{2}{16807} a^{27} + \frac{20}{16807} a^{25} + \frac{5}{16807} a^{23} + \frac{22}{16807} a^{21} - \frac{23}{16807} a^{19} + \frac{92}{16807} a^{17} - \frac{146}{16807} a^{15} - \frac{935}{16807} a^{13} + \frac{1002}{16807} a^{11} + \frac{516}{16807} a^{9} + \frac{822}{16807} a^{7} - \frac{800}{16807} a^{5} - \frac{2}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{16807} a^{32} - \frac{2}{16807} a^{28} - \frac{1}{16807} a^{26} - \frac{23}{16807} a^{24} + \frac{1}{16807} a^{22} - \frac{135}{16807} a^{20} + \frac{57}{16807} a^{18} + \frac{85}{16807} a^{16} - \frac{67}{16807} a^{14} + \frac{673}{16807} a^{12} - \frac{1017}{16807} a^{10} + \frac{66}{16807} a^{8} - \frac{8325}{16807} a^{6} + \frac{88}{2401} a^{4} - \frac{19}{49} a^{2}$, $\frac{1}{117649} a^{33} - \frac{1}{117649} a^{31} + \frac{1}{117649} a^{29} - \frac{3}{117649} a^{27} + \frac{101}{117649} a^{25} + \frac{144}{117649} a^{23} - \frac{113}{117649} a^{21} - \frac{377}{117649} a^{19} + \frac{83}{117649} a^{17} - \frac{311}{117649} a^{15} + \frac{997}{117649} a^{13} + \frac{4883}{117649} a^{11} + \frac{1758}{117649} a^{9} - \frac{3326}{117649} a^{7} + \frac{36156}{117649} a^{5} - \frac{597}{2401} a^{3} + \frac{11}{49} a$, $\frac{1}{38689226597} a^{34} + \frac{943480}{38689226597} a^{32} - \frac{135295}{38689226597} a^{30} + \frac{3898752}{38689226597} a^{28} - \frac{56477}{354947033} a^{26} - \frac{37142206}{38689226597} a^{24} + \frac{38440940}{38689226597} a^{22} - \frac{177667027}{38689226597} a^{20} - \frac{72606969}{38689226597} a^{18} - \frac{321057166}{38689226597} a^{16} - \frac{286267853}{38689226597} a^{14} - \frac{450268136}{38689226597} a^{12} + \frac{1616676958}{38689226597} a^{10} - \frac{161463359}{38689226597} a^{8} + \frac{15372658435}{38689226597} a^{6} - \frac{360996155}{789576053} a^{4} + \frac{2802321}{16113797} a^{2} + \frac{19496}{46979}$, $\frac{1}{270824586179} a^{35} + \frac{943480}{270824586179} a^{33} - \frac{135295}{270824586179} a^{31} - \frac{705190}{270824586179} a^{29} + \frac{196951}{2484629231} a^{27} - \frac{129221046}{270824586179} a^{25} - \frac{242399522}{270824586179} a^{23} - \frac{255934041}{270824586179} a^{21} - \frac{1315671309}{270824586179} a^{19} - \frac{1184296291}{270824586179} a^{17} + \frac{2294241638}{270824586179} a^{15} - \frac{13263038722}{270824586179} a^{13} + \frac{17633791176}{270824586179} a^{11} - \frac{16312091895}{270824586179} a^{9} - \frac{3395311128}{270824586179} a^{7} + \frac{1589243072}{5527032371} a^{5} + \frac{33432629}{112796579} a^{3} - \frac{286339}{2301971} a$, $\frac{1}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{36} - \frac{4581726622875218810992401735645417235059090673040383638436311261519}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{34} + \frac{11873232812594360852180481300466411981091984122532769347423194503341655668}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{32} + \frac{6683277042913664224038591182489480796579705134699936066083647553360462656}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{30} - \frac{13767634298325273025982615489841217765581890239455423865151298291682330611}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{28} + \frac{53266941091159648042027264595544376551073137274582048190644303665350238981}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{26} - \frac{152078066337504608331051249487192002982086079013403186305631162455155120882}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{24} - \frac{291188423880634312258985131210123151421016138499042735958574444018041931250}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{22} + \frac{2891914232731322939549414694547201465590747969182404228173889463582380142206}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{20} - \frac{1100387562717850943059930924188652812243958923860927150790060361838688369872}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{18} - \frac{3654928535513055671801351650056811120861147813581712161363991763064445821356}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{16} + \frac{3198078697050649509174194387109024077741348061814458172570960640275617581519}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{14} + \frac{19711354082735511273826386651435018556078884554062426857419503888523761502074}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{12} + \frac{20010584963179492726320163537637721582402033316142201219271063989611126668566}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{10} - \frac{27317850157054000973467969892611874089076844946314550243135249059023107623854}{477930393417471341536692345445681211783330834731663242098639874480095336336239} a^{8} + \frac{2507997335166193359842711159367032044124230368493378016356554325880061457370}{9753681498315741664014129498891453301700629280238025348951834173063170129311} a^{6} + \frac{56016028198842813839377004389391371824437247573149629591267013461214266237}{199054724455423299265594479569213332687767944494653578550037432103330002639} a^{4} - \frac{924685338474510932655223846365736391967627525687108584962589869868204550}{4062341315416802025828458766718639442607509071319460786735457798027142911} a^{2} - \frac{521823032927252126904687878983954515681679808489965665039755090112775}{1691937240906623084476659211461324215996463586555377253950627987516511}$, $\frac{1}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{37} - \frac{4581726622875218810992401735645417235059090673040383638436311261519}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{35} + \frac{11873232812594360852180481300466411981091984122532769347423194503341655668}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{33} - \frac{50189501372921564137559831551571471399925421863772514948212761619019538098}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{31} - \frac{42204023506242887206781826856871693863834453738691649372299502877872330988}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{29} + \frac{536685557625759089115613857835062470221366716761597881812163781630580245390}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{27} + \frac{729449999107941431273724302890752756063743389462919804415963179716734890805}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{25} + \frac{4855798022752453854465672126222393022362697854862714080835250586082348136987}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{23} - \frac{2425690549149270912260037831087497564782481405174769941662824794035149928293}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{21} + \frac{3079761650846038341517553146764827174199167910526797998760725712331241685547}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{19} + \frac{9368937721713211623004687156043146932138526269068479120909885937410574351310}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{17} + \frac{23160423921008814664095240766764418298714647638278288478589000259780997846173}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{15} + \frac{192405545742419182193819997283411099900766702684924024362330550340455633791595}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{13} + \frac{107139681496239062576288947166219100347447887877801996173173162841697287823694}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{11} - \frac{104664828802589911545241824810934769076323817664237083622578365533459908649294}{3345512753922299390756846418119768482483315843121642694690479121360667354353673} a^{9} - \frac{4583689932604281033817826859104649954256306896010023528487344858675893738690}{68275770488210191648098906492240173111904404961666177442662839211442190905177} a^{7} - \frac{491878932712226219737446118080494865069049566270318520423802995852294941360}{1393383071187963094859161356984493328814375611462575049850262024723310018473} a^{5} + \frac{8313291996875651108587335448213093827373063656905251221607838941971945510}{28436389207917614180799211367030476098252563499236225507148204586190000377} a^{3} - \frac{3663992194610980712361340700269270916817969326378523422376635638357724}{11843560686346361591336614480229269511975245105887640777654395912615577} a$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $18$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{33308191601483491905535406226328031558634182}{610446157838708770031745137371927608945705071238955009} a^{37} + \frac{5981483706916529413575589164019337752868021948}{610446157838708770031745137371927608945705071238955009} a^{35} + \frac{467882443534065234970454486004005035638674135008}{610446157838708770031745137371927608945705071238955009} a^{33} + \frac{21179044688584257349886137367099326328585694304178}{610446157838708770031745137371927608945705071238955009} a^{31} + \frac{621626333758176364437576241063501395625411873572332}{610446157838708770031745137371927608945705071238955009} a^{29} + \frac{12558135516279590172583951574497871126696589482496134}{610446157838708770031745137371927608945705071238955009} a^{27} + \frac{180756588571772554949260906124708455441348053839215435}{610446157838708770031745137371927608945705071238955009} a^{25} + \frac{1890901367320887279912702126443139481230970735117715935}{610446157838708770031745137371927608945705071238955009} a^{23} + \frac{14516231308724617503345481857384256606642459284598587005}{610446157838708770031745137371927608945705071238955009} a^{21} + \frac{81894333304294757008849118694862209435414892722436933375}{610446157838708770031745137371927608945705071238955009} a^{19} + \frac{337347215640377417883931140592774849041961250757068133830}{610446157838708770031745137371927608945705071238955009} a^{17} + \frac{999585628850362205337815284027206584402311005280207105308}{610446157838708770031745137371927608945705071238955009} a^{15} + \frac{2076191111983259095288008401323425162962864335343492072262}{610446157838708770031745137371927608945705071238955009} a^{13} + \frac{2898661064823391776355309822859346890617237200897457186215}{610446157838708770031745137371927608945705071238955009} a^{11} + \frac{2534500073478219301847685979923303126541749379317650999367}{610446157838708770031745137371927608945705071238955009} a^{9} + \frac{24694441415857256482185950187627452843712220531568353081}{12458084853851199388402961987182196100932756555897041} a^{7} + \frac{87906080776512496022928918031762481720062045639197398}{254246629670432640579652285452697879610872582773409} a^{5} - \frac{196855514226921362214023609427636702384300475363096}{5188706727968013073054128274544854685936175158641} a^{3} - \frac{19611055610337039773886731863037116011909412268}{2161060694697214940880519897769618777982580241} a \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_{38}$ (as 38T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 38
The 38 conjugacy class representatives for $C_{38}$
Character table for $C_{38}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 19.19.114445997944945591651333831028437092270721.1

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 R $38$ $19^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{19}$ $38$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $38$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $38$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
191Data not computed