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Properties

Label 40.0.15046568208...9961.1
Degree $40$
Signature $[0, 20]$
Discriminant $13^{20}\cdot 41^{39}$
Root discriminant $134.72$
Ramified primes $13, 41$
Class number Not computed
Class group Not computed
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![711732296356909, -711589338196468, 711589338196468, -708253647786178, 708253647786178, -685015004594491, 685015004594491, -608659462678948, 608659462678948, -464432327949589, 464432327949589, -289611558580669, 289611558580669, -143927584106569, 143927584106569, -56517199422109, 56517199422109, -17525312136394, 17525312136394, -4299993641824, 4299993641824, -836219750389, 836219750389, -128861445814, 128861445814, -15684117082, 15684117082, -1496645674, 1496645674, -110513410, 110513410, -6180874, 6180874, -252889, 252889, -7135, 7135, -124, 124, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 124*x^38 - 124*x^37 + 7135*x^36 - 7135*x^35 + 252889*x^34 - 252889*x^33 + 6180874*x^32 - 6180874*x^31 + 110513410*x^30 - 110513410*x^29 + 1496645674*x^28 - 1496645674*x^27 + 15684117082*x^26 - 15684117082*x^25 + 128861445814*x^24 - 128861445814*x^23 + 836219750389*x^22 - 836219750389*x^21 + 4299993641824*x^20 - 4299993641824*x^19 + 17525312136394*x^18 - 17525312136394*x^17 + 56517199422109*x^16 - 56517199422109*x^15 + 143927584106569*x^14 - 143927584106569*x^13 + 289611558580669*x^12 - 289611558580669*x^11 + 464432327949589*x^10 - 464432327949589*x^9 + 608659462678948*x^8 - 608659462678948*x^7 + 685015004594491*x^6 - 685015004594491*x^5 + 708253647786178*x^4 - 708253647786178*x^3 + 711589338196468*x^2 - 711589338196468*x + 711732296356909)
 
gp: K = bnfinit(x^40 - x^39 + 124*x^38 - 124*x^37 + 7135*x^36 - 7135*x^35 + 252889*x^34 - 252889*x^33 + 6180874*x^32 - 6180874*x^31 + 110513410*x^30 - 110513410*x^29 + 1496645674*x^28 - 1496645674*x^27 + 15684117082*x^26 - 15684117082*x^25 + 128861445814*x^24 - 128861445814*x^23 + 836219750389*x^22 - 836219750389*x^21 + 4299993641824*x^20 - 4299993641824*x^19 + 17525312136394*x^18 - 17525312136394*x^17 + 56517199422109*x^16 - 56517199422109*x^15 + 143927584106569*x^14 - 143927584106569*x^13 + 289611558580669*x^12 - 289611558580669*x^11 + 464432327949589*x^10 - 464432327949589*x^9 + 608659462678948*x^8 - 608659462678948*x^7 + 685015004594491*x^6 - 685015004594491*x^5 + 708253647786178*x^4 - 708253647786178*x^3 + 711589338196468*x^2 - 711589338196468*x + 711732296356909, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 124 x^{38} - 124 x^{37} + 7135 x^{36} - 7135 x^{35} + 252889 x^{34} - 252889 x^{33} + 6180874 x^{32} - 6180874 x^{31} + 110513410 x^{30} - 110513410 x^{29} + 1496645674 x^{28} - 1496645674 x^{27} + 15684117082 x^{26} - 15684117082 x^{25} + 128861445814 x^{24} - 128861445814 x^{23} + 836219750389 x^{22} - 836219750389 x^{21} + 4299993641824 x^{20} - 4299993641824 x^{19} + 17525312136394 x^{18} - 17525312136394 x^{17} + 56517199422109 x^{16} - 56517199422109 x^{15} + 143927584106569 x^{14} - 143927584106569 x^{13} + 289611558580669 x^{12} - 289611558580669 x^{11} + 464432327949589 x^{10} - 464432327949589 x^{9} + 608659462678948 x^{8} - 608659462678948 x^{7} + 685015004594491 x^{6} - 685015004594491 x^{5} + 708253647786178 x^{4} - 708253647786178 x^{3} + 711589338196468 x^{2} - 711589338196468 x + 711732296356909 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15046568208788745095239372086809585688465158910040879547984296481209340598707851489961=13^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.72$
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:  $13, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(533=13\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{533}(1,·)$, $\chi_{533}(259,·)$, $\chi_{533}(519,·)$, $\chi_{533}(12,·)$, $\chi_{533}(66,·)$, $\chi_{533}(142,·)$, $\chi_{533}(144,·)$, $\chi_{533}(131,·)$, $\chi_{533}(376,·)$, $\chi_{533}(326,·)$, $\chi_{533}(38,·)$, $\chi_{533}(40,·)$, $\chi_{533}(92,·)$, $\chi_{533}(298,·)$, $\chi_{533}(430,·)$, $\chi_{533}(181,·)$, $\chi_{533}(311,·)$, $\chi_{533}(443,·)$, $\chi_{533}(194,·)$, $\chi_{533}(196,·)$, $\chi_{533}(454,·)$, $\chi_{533}(456,·)$, $\chi_{533}(129,·)$, $\chi_{533}(209,·)$, $\chi_{533}(469,·)$, $\chi_{533}(220,·)$, $\chi_{533}(378,·)$, $\chi_{533}(350,·)$, $\chi_{533}(480,·)$, $\chi_{533}(272,·)$, $\chi_{533}(482,·)$, $\chi_{533}(105,·)$, $\chi_{533}(363,·)$, $\chi_{533}(365,·)$, $\chi_{533}(116,·)$, $\chi_{533}(118,·)$, $\chi_{533}(233,·)$, $\chi_{533}(248,·)$, $\chi_{533}(506,·)$, $\chi_{533}(508,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{197399022252931} a^{21} - \frac{2425063538251}{197399022252931} a^{20} + \frac{63}{197399022252931} a^{19} + \frac{51895209957871}{197399022252931} a^{18} + \frac{1701}{197399022252931} a^{17} + \frac{40234209281659}{197399022252931} a^{16} + \frac{25704}{197399022252931} a^{15} - \frac{70631529194885}{197399022252931} a^{14} + \frac{238140}{197399022252931} a^{13} + \frac{32802869432759}{197399022252931} a^{12} + \frac{1393119}{197399022252931} a^{11} - \frac{8407949867729}{197399022252931} a^{10} + \frac{5108103}{197399022252931} a^{9} - \frac{97525203950890}{197399022252931} a^{8} + \frac{11258676}{197399022252931} a^{7} - \frac{28201128637850}{197399022252931} a^{6} + \frac{13640319}{197399022252931} a^{5} - \frac{1763680316368}{197399022252931} a^{4} + \frac{7577955}{197399022252931} a^{3} + \frac{53194758681211}{197399022252931} a^{2} + \frac{1240029}{197399022252931} a + \frac{30827548636283}{197399022252931}$, $\frac{1}{197399022252931} a^{22} + \frac{66}{197399022252931} a^{20} + \frac{7275190614753}{197399022252931} a^{19} + \frac{1881}{197399022252931} a^{18} + \frac{19887820535059}{197399022252931} a^{17} + \frac{30294}{197399022252931} a^{16} + \frac{82509648335554}{197399022252931} a^{15} + \frac{302940}{197399022252931} a^{14} - \frac{52105243550207}{197399022252931} a^{13} + \frac{1945944}{197399022252931} a^{12} - \frac{90582464086925}{197399022252931} a^{11} + \frac{8027019}{197399022252931} a^{10} - \frac{4033711582080}{197399022252931} a^{9} + \frac{20640906}{197399022252931} a^{8} - \frac{71908438920508}{197399022252931} a^{7} + \frac{30961359}{197399022252931} a^{6} + \frac{89536363872169}{197399022252931} a^{5} + \frac{23816430}{197399022252931} a^{4} + \frac{16184106673540}{197399022252931} a^{3} + \frac{7144929}{197399022252931} a^{2} + \frac{3236821334708}{197399022252931} a + \frac{354294}{197399022252931}$, $\frac{1}{197399022252931} a^{23} - \frac{30069638113612}{197399022252931} a^{20} - \frac{2277}{197399022252931} a^{19} - \frac{49412658384600}{197399022252931} a^{18} - \frac{81972}{197399022252931} a^{17} - \frac{6760874965837}{197399022252931} a^{16} - \frac{1393524}{197399022252931} a^{15} + \frac{69398171494790}{197399022252931} a^{14} - \frac{13771296}{197399022252931} a^{13} - \frac{84182601866778}{197399022252931} a^{12} - \frac{83918835}{197399022252931} a^{11} - \frac{41306087070759}{197399022252931} a^{10} - \frac{316493892}{197399022252931} a^{9} + \frac{47986309744440}{197399022252931} a^{8} - \frac{712111257}{197399022252931} a^{7} - \frac{23179368559041}{197399022252931} a^{6} - \frac{876444624}{197399022252931} a^{5} - \frac{64812014699103}{197399022252931} a^{4} - \frac{493000101}{197399022252931} a^{3} + \frac{45565148927540}{197399022252931} a^{2} - \frac{81487620}{197399022252931} a - \frac{60627987465368}{197399022252931}$, $\frac{1}{197399022252931} a^{24} - \frac{2484}{197399022252931} a^{20} + \frac{68383342496577}{197399022252931} a^{19} - \frac{94392}{197399022252931} a^{18} + \frac{15346792779046}{197399022252931} a^{17} - \frac{1710234}{197399022252931} a^{16} - \frac{35194898700158}{197399022252931} a^{15} - \frac{18242496}{197399022252931} a^{14} + \frac{49905548622877}{197399022252931} a^{13} - \frac{122063760}{197399022252931} a^{12} - \frac{95836251281234}{197399022252931} a^{11} - \frac{517899096}{197399022252931} a^{10} - \frac{86157951609658}{197399022252931} a^{9} - \frac{1359485127}{197399022252931} a^{8} + \frac{31639707150396}{197399022252931} a^{7} - \frac{2071596384}{197399022252931} a^{6} - \frac{47746351614364}{197399022252931} a^{5} - \frac{1613454876}{197399022252931} a^{4} + \frac{33112848168736}{197399022252931} a^{3} - \frac{488925720}{197399022252931} a^{2} + \frac{66540996066928}{197399022252931} a - \frac{24446286}{197399022252931}$, $\frac{1}{197399022252931} a^{25} - \frac{33503818930977}{197399022252931} a^{20} + \frac{62100}{197399022252931} a^{19} + \frac{21486796966667}{197399022252931} a^{18} + \frac{2515050}{197399022252931} a^{17} + \frac{22675696957712}{197399022252931} a^{16} + \frac{45606240}{197399022252931} a^{15} + \frac{88917811384196}{197399022252931} a^{14} + \frac{469476000}{197399022252931} a^{13} + \frac{58094251484550}{197399022252931} a^{12} + \frac{2942608500}{197399022252931} a^{11} - \frac{47209064237808}{197399022252931} a^{10} + \frac{11329042725}{197399022252931} a^{9} - \frac{12366602514027}{197399022252931} a^{8} + \frac{25894954800}{197399022252931} a^{7} - \frac{22696988243259}{197399022252931} a^{6} + \frac{32269097520}{197399022252931} a^{5} - \frac{5090568124894}{197399022252931} a^{4} + \frac{18334714500}{197399022252931} a^{3} - \frac{55023349268718}{197399022252931} a^{2} + \frac{3055785750}{197399022252931} a - \frac{15189821610256}{197399022252931}$, $\frac{1}{197399022252931} a^{26} + \frac{70200}{197399022252931} a^{20} - \frac{39161855164023}{197399022252931} a^{19} + \frac{3001050}{197399022252931} a^{18} - \frac{35645732547470}{197399022252931} a^{17} + \frac{57999240}{197399022252931} a^{16} + \frac{19145523679051}{197399022252931} a^{15} + \frac{644436000}{197399022252931} a^{14} - \frac{13545966870759}{197399022252931} a^{13} + \frac{4435236000}{197399022252931} a^{12} - \frac{41896443772564}{197399022252931} a^{11} + \frac{19210115925}{197399022252931} a^{10} - \frac{58686668225776}{197399022252931} a^{9} + \frac{51226975800}{197399022252931} a^{8} + \frac{12180555832879}{197399022252931} a^{7} + \frac{79035905520}{197399022252931} a^{6} - \frac{46350052985813}{197399022252931} a^{5} + \frac{62178597000}{197399022252931} a^{4} + \frac{97520512255599}{197399022252931} a^{3} + \frac{18999015750}{197399022252931} a^{2} - \frac{90722146507769}{197399022252931} a + \frac{956593800}{197399022252931}$, $\frac{1}{197399022252931} a^{27} + \frac{42341348029655}{197399022252931} a^{20} - \frac{1421550}{197399022252931} a^{19} - \frac{80429097250065}{197399022252931} a^{18} - \frac{61410960}{197399022252931} a^{17} - \frac{37135653846001}{197399022252931} a^{16} - \frac{1159984800}{197399022252931} a^{15} + \frac{51162564935383}{197399022252931} a^{14} - \frac{12282192000}{197399022252931} a^{13} + \frac{53662979238682}{197399022252931} a^{12} - \frac{78586837875}{197399022252931} a^{11} - \frac{43682489913666}{197399022252931} a^{10} - \frac{307361854800}{197399022252931} a^{9} + \frac{88608132157937}{197399022252931} a^{8} - \frac{711323149680}{197399022252931} a^{7} - \frac{41913850560812}{197399022252931} a^{6} - \frac{895371796800}{197399022252931} a^{5} - \frac{58707253551469}{197399022252931} a^{4} - \frac{512973425250}{197399022252931} a^{3} + \frac{31921413428689}{197399022252931} a^{2} - \frac{86093442000}{197399022252931} a - \frac{8433308184047}{197399022252931}$, $\frac{1}{197399022252931} a^{28} - \frac{1658475}{197399022252931} a^{20} + \frac{15652288422704}{197399022252931} a^{19} - \frac{75626460}{197399022252931} a^{18} - \frac{9125529969341}{197399022252931} a^{17} - \frac{1522480050}{197399022252931} a^{16} - \frac{30037508908134}{197399022252931} a^{15} - \frac{17399772000}{197399022252931} a^{14} + \frac{27099876912462}{197399022252931} a^{13} - \frac{122246192250}{197399022252931} a^{12} - \frac{1677616269122}{197399022252931} a^{11} - \frac{537883245900}{197399022252931} a^{10} - \frac{86372368233620}{197399022252931} a^{9} - \frac{1452284763930}{197399022252931} a^{8} + \frac{23612607603272}{197399022252931} a^{7} - \frac{2263300930800}{197399022252931} a^{6} - \frac{85611210450407}{197399022252931} a^{5} - \frac{1795406988375}{197399022252931} a^{4} + \frac{60841220096597}{197399022252931} a^{3} - \frac{552432919500}{197399022252931} a^{2} + \frac{78847705849200}{197399022252931} a - \frac{27980368650}{197399022252931}$, $\frac{1}{197399022252931} a^{29} - \frac{83919931188327}{197399022252931} a^{20} + \frac{28857465}{197399022252931} a^{19} - \frac{61488039043271}{197399022252931} a^{18} + \frac{1298585925}{197399022252931} a^{17} + \frac{16511665477168}{197399022252931} a^{16} + \frac{25229669400}{197399022252931} a^{15} - \frac{70496275673893}{197399022252931} a^{14} + \frac{272703044250}{197399022252931} a^{13} - \frac{38529984564335}{197399022252931} a^{12} + \frac{1772569787625}{197399022252931} a^{11} + \frac{3301719211876}{197399022252931} a^{10} + \frac{7019376358995}{197399022252931} a^{9} - \frac{54747438363077}{197399022252931} a^{8} + \frac{16408931748300}{197399022252931} a^{7} - \frac{17692348269741}{197399022252931} a^{6} + \frac{20826721065150}{197399022252931} a^{5} - \frac{97558746643576}{197399022252931} a^{4} + \frac{12015415999125}{197399022252931} a^{3} + \frac{93029180581112}{197399022252931} a^{2} + \frac{2028576727125}{197399022252931} a - \frac{22836994186437}{197399022252931}$, $\frac{1}{197399022252931} a^{30} + \frac{34628958}{197399022252931} a^{20} + \frac{93093047245124}{197399022252931} a^{19} + \frac{1644875505}{197399022252931} a^{18} + \frac{44821527952282}{197399022252931} a^{17} + \frac{34060053690}{197399022252931} a^{16} + \frac{28298831306278}{197399022252931} a^{15} + \frac{397367293050}{197399022252931} a^{14} - \frac{23129683106995}{197399022252931} a^{13} + \frac{2836111660200}{197399022252931} a^{12} + \frac{93393482737315}{197399022252931} a^{11} + \frac{12634877446191}{197399022252931} a^{10} + \frac{78190005643935}{197399022252931} a^{9} + \frac{34458756671430}{197399022252931} a^{8} - \frac{26828730233262}{197399022252931} a^{7} + \frac{54149474769390}{197399022252931} a^{6} - \frac{89647149575060}{197399022252931} a^{5} + \frac{43255497596850}{197399022252931} a^{4} + \frac{75491287591073}{197399022252931} a^{3} + \frac{13388606399025}{197399022252931} a^{2} + \frac{85554435866845}{197399022252931} a + \frac{681601780314}{197399022252931}$, $\frac{1}{197399022252931} a^{31} + \frac{24459630611562}{197399022252931} a^{20} - \frac{536748849}{197399022252931} a^{19} + \frac{72195997250113}{197399022252931} a^{18} - \frac{24843803868}{197399022252931} a^{17} + \frac{35451220515710}{197399022252931} a^{16} - \frac{492735443382}{197399022252931} a^{15} - \frac{38271848534385}{197399022252931} a^{14} - \frac{5410428397920}{197399022252931} a^{13} + \frac{25899078388935}{197399022252931} a^{12} - \frac{35607381893811}{197399022252931} a^{11} - \frac{1821818161408}{197399022252931} a^{10} + \frac{54969494677687}{197399022252931} a^{9} - \frac{72510640050867}{197399022252931} a^{8} + \frac{59071300935644}{197399022252931} a^{7} + \frac{8231498995144}{197399022252931} a^{6} - \frac{34296491654890}{197399022252931} a^{5} + \frac{19203253935941}{197399022252931} a^{4} - \frac{51629056768934}{197399022252931} a^{3} - \frac{60531499623250}{197399022252931} a^{2} - \frac{42259310379468}{197399022252931} a - \frac{67984862805285}{197399022252931}$, $\frac{1}{197399022252931} a^{32} - \frac{660613968}{197399022252931} a^{20} - \frac{86967575507776}{197399022252931} a^{19} - \frac{32275711008}{197399022252931} a^{18} + \frac{80813245617189}{197399022252931} a^{17} - \frac{682249075452}{197399022252931} a^{16} - \frac{32731212538798}{197399022252931} a^{15} - \frac{8085914968320}{197399022252931} a^{14} + \frac{59813880502203}{197399022252931} a^{13} - \frac{58432626697536}{197399022252931} a^{12} + \frac{38660556398865}{197399022252931} a^{11} - \frac{65547797885981}{197399022252931} a^{10} - \frac{55674614831820}{197399022252931} a^{9} + \frac{66492333629716}{197399022252931} a^{8} + \frac{37282369998230}{197399022252931} a^{7} + \frac{40142037088914}{197399022252931} a^{6} - \frac{29115469455791}{197399022252931} a^{5} + \frac{67506819491615}{197399022252931} a^{4} + \frac{88090562717282}{197399022252931} a^{3} - \frac{88664001854237}{197399022252931} a^{2} + \frac{35294719933429}{197399022252931} a - \frac{14628222823662}{197399022252931}$, $\frac{1}{197399022252931} a^{33} - \frac{83523620786906}{197399022252931} a^{20} + \frac{9342968976}{197399022252931} a^{19} + \frac{90051506616458}{197399022252931} a^{18} + \frac{441455284116}{197399022252931} a^{17} - \frac{93254090436835}{197399022252931} a^{16} + \frac{8894506465152}{197399022252931} a^{15} + \frac{29804461939216}{197399022252931} a^{14} - \frac{98513038610947}{197399022252931} a^{13} + \frac{35560201544950}{197399022252931} a^{12} + \frac{65169983588487}{197399022252931} a^{11} - \frac{32564684009905}{197399022252931} a^{10} + \frac{85193147112593}{197399022252931} a^{9} + \frac{12864779370777}{197399022252931} a^{8} - \frac{23382181736096}{197399022252931} a^{7} + \frac{62466763253248}{197399022252931} a^{6} - \frac{1862944767419}{197399022252931} a^{5} + \frac{38237078083116}{197399022252931} a^{4} - \frac{17536636302072}{197399022252931} a^{3} + \frac{88947642703889}{197399022252931} a^{2} + \frac{14956166289686}{197399022252931} a + \frac{82178784267400}{197399022252931}$, $\frac{1}{197399022252931} a^{34} + \frac{11765220192}{197399022252931} a^{20} + \frac{22266015362399}{197399022252931} a^{19} + \frac{586790357076}{197399022252931} a^{18} + \frac{50527868232882}{197399022252931} a^{17} + \frac{12600550825632}{197399022252931} a^{16} + \frac{9187145693484}{197399022252931} a^{15} - \frac{46192412345347}{197399022252931} a^{14} + \frac{30334145506368}{197399022252931} a^{13} - \frac{80666740341906}{197399022252931} a^{12} + \frac{75042203741442}{197399022252931} a^{11} + \frac{73187490211373}{197399022252931} a^{10} + \frac{78424997370831}{197399022252931} a^{9} + \frac{50827648083086}{197399022252931} a^{8} - \frac{66974664689476}{197399022252931} a^{7} - \frac{31890123930640}{197399022252931} a^{6} + \frac{18074671173768}{197399022252931} a^{5} + \frac{65349833249682}{197399022252931} a^{4} + \frac{476971199583}{1211036946337} a^{3} + \frac{45096700672142}{197399022252931} a^{2} + \frac{80345838184732}{197399022252931} a + \frac{88664001854237}{197399022252931}$, $\frac{1}{197399022252931} a^{35} - \frac{86068945440936}{197399022252931} a^{20} - \frac{154418515020}{197399022252931} a^{19} + \frac{64597954379505}{197399022252931} a^{18} - \frac{7412088720960}{197399022252931} a^{17} + \frac{71875809052974}{197399022252931} a^{16} + \frac{46192412345347}{197399022252931} a^{15} - \frac{36809420285558}{197399022252931} a^{14} + \frac{78549056929179}{197399022252931} a^{13} + \frac{8058652602209}{197399022252931} a^{12} + \frac{66954548545798}{197399022252931} a^{11} - \frac{69953169501271}{197399022252931} a^{10} - \frac{37826145441666}{197399022252931} a^{9} + \frac{76492440302265}{197399022252931} a^{8} - \frac{37948402599731}{197399022252931} a^{7} - \frac{5569244441917}{197399022252931} a^{6} + \frac{69398400761337}{197399022252931} a^{5} - \frac{30781937593333}{197399022252931} a^{4} - \frac{84253443323337}{197399022252931} a^{3} - \frac{66791171453362}{197399022252931} a^{2} - \frac{90421603147368}{197399022252931} a + \frac{78027349213004}{197399022252931}$, $\frac{1}{197399022252931} a^{36} - \frac{198538090740}{197399022252931} a^{20} - \frac{40231105923595}{197399022252931} a^{19} - \frac{10059263264160}{197399022252931} a^{18} + \frac{5077492410308}{197399022252931} a^{17} - \frac{21310538506253}{197399022252931} a^{16} + \frac{28521804935669}{197399022252931} a^{15} - \frac{84837689611097}{197399022252931} a^{14} - \frac{65951631483274}{197399022252931} a^{13} + \frac{30179645112969}{197399022252931} a^{12} - \frac{98245335782838}{197399022252931} a^{11} + \frac{68048878257452}{197399022252931} a^{10} + \frac{38554171354163}{197399022252931} a^{9} - \frac{36034150286162}{197399022252931} a^{8} + \frac{39725360980507}{197399022252931} a^{7} - \frac{18174836007273}{197399022252931} a^{6} + \frac{54463299322747}{197399022252931} a^{5} - \frac{42978963885827}{197399022252931} a^{4} + \frac{24433742889556}{197399022252931} a^{3} - \frac{75412701533728}{197399022252931} a^{2} - \frac{57786009069484}{197399022252931} a - \frac{78059074804354}{197399022252931}$, $\frac{1}{197399022252931} a^{37} - \frac{62694475988545}{197399022252931} a^{20} + \frac{2448636452460}{197399022252931} a^{19} + \frac{39068444283825}{197399022252931} a^{18} - \frac{78395290663375}{197399022252931} a^{17} + \frac{6995477572163}{197399022252931} a^{16} + \frac{83409838446588}{197399022252931} a^{15} - \frac{67948336698876}{197399022252931} a^{14} - \frac{65724766766871}{197399022252931} a^{13} - \frac{78267410481018}{197399022252931} a^{12} - \frac{98193886733750}{197399022252931} a^{11} + \frac{25598203596014}{197399022252931} a^{10} + \frac{78205459673511}{197399022252931} a^{9} + \frac{12181109061666}{197399022252931} a^{8} - \frac{88665527937677}{197399022252931} a^{7} + \frac{33981453049953}{197399022252931} a^{6} - \frac{37273907300156}{197399022252931} a^{5} + \frac{29906889790428}{197399022252931} a^{4} + \frac{59356122515821}{197399022252931} a^{3} - \frac{91686505151518}{197399022252931} a^{2} - \frac{41649701977851}{197399022252931} a - \frac{33617873325391}{197399022252931}$, $\frac{1}{197399022252931} a^{38} + \frac{3208558110120}{197399022252931} a^{20} + \frac{40839986503540}{197399022252931} a^{19} - \frac{32799991203775}{197399022252931} a^{18} + \frac{54827117504468}{197399022252931} a^{17} + \frac{61695841148802}{197399022252931} a^{16} + \frac{62643822186051}{197399022252931} a^{15} - \frac{35535808304144}{197399022252931} a^{14} - \frac{93404576557972}{197399022252931} a^{13} - \frac{19537198765329}{197399022252931} a^{12} - \frac{82694120179460}{197399022252931} a^{11} + \frac{3963196959505}{197399022252931} a^{10} - \frac{55892374360187}{197399022252931} a^{9} + \frac{11096951486614}{197399022252931} a^{8} - \frac{31458480465324}{197399022252931} a^{7} - \frac{80865124808693}{197399022252931} a^{6} + \frac{45527269499290}{197399022252931} a^{5} - \frac{18522960570857}{197399022252931} a^{4} + \frac{4724382354708}{197399022252931} a^{3} + \frac{19757459741962}{197399022252931} a^{2} + \frac{93419720809098}{197399022252931} a - \frac{71327198752925}{197399022252931}$, $\frac{1}{197399022252931} a^{39} - \frac{86225421488568}{197399022252931} a^{20} - \frac{37540129888404}{197399022252931} a^{19} - \frac{67934516410634}{197399022252931} a^{18} - \frac{66287903336181}{197399022252931} a^{17} + \frac{90606513397846}{197399022252931} a^{16} + \frac{4477830896534}{197399022252931} a^{15} - \frac{70211801338148}{197399022252931} a^{14} + \frac{26049598353772}{197399022252931} a^{13} + \frac{16469732288198}{197399022252931} a^{12} + \frac{4157280064789}{197399022252931} a^{11} + \frac{64998761273068}{197399022252931} a^{10} + \frac{11808589539322}{197399022252931} a^{9} + \frac{89221496726875}{197399022252931} a^{8} + \frac{21417170416118}{197399022252931} a^{7} + \frac{86106591466798}{197399022252931} a^{6} - \frac{42653292661265}{197399022252931} a^{5} - \frac{6582258863986}{197399022252931} a^{4} - \frac{59447954392575}{197399022252931} a^{3} + \frac{35472117933319}{197399022252931} a^{2} - \frac{1739402669169}{197399022252931} a - \frac{94280406408895}{197399022252931}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 5.5.2825761.1, 8.0.5562376816315241.1, 10.10.327381934393961.1, \(\Q(\zeta_{41})^+\)

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 $20^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ $20^{2}$ $40$ $40$ R $40$ $40$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
13Data not computed
41Data not computed