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Properties

Label 40.40.5326281494...4361.1
Degree $40$
Signature $[40, 0]$
Discriminant $11^{20}\cdot 41^{39}$
Root discriminant $123.93$
Ramified primes $11, 41$
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![-6414099449, 149372259890, -149372259890, -3186318150400, 3186318150400, 20052325041287, -20052325041287, -56303216874256, 56303216874256, 87923917855103, -87923917855103, -86896851513817, 86896851513817, 58787122960283, -58787122960283, -28623261724177, 28623261724177, 10368625561538, -10368625561538, -2856692933032, 2856692933032, 607080958403, -607080958403, -100277346172, 100277346172, 12899982560, -12899982560, -1287488848, 1287488848, 98643416, -98643416, -5689120, 5689120, 238865, -238865, -6889, 6889, 122, -122, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 122*x^38 + 122*x^37 + 6889*x^36 - 6889*x^35 - 238865*x^34 + 238865*x^33 + 5689120*x^32 - 5689120*x^31 - 98643416*x^30 + 98643416*x^29 + 1287488848*x^28 - 1287488848*x^27 - 12899982560*x^26 + 12899982560*x^25 + 100277346172*x^24 - 100277346172*x^23 - 607080958403*x^22 + 607080958403*x^21 + 2856692933032*x^20 - 2856692933032*x^19 - 10368625561538*x^18 + 10368625561538*x^17 + 28623261724177*x^16 - 28623261724177*x^15 - 58787122960283*x^14 + 58787122960283*x^13 + 86896851513817*x^12 - 86896851513817*x^11 - 87923917855103*x^10 + 87923917855103*x^9 + 56303216874256*x^8 - 56303216874256*x^7 - 20052325041287*x^6 + 20052325041287*x^5 + 3186318150400*x^4 - 3186318150400*x^3 - 149372259890*x^2 + 149372259890*x - 6414099449)
 
gp: K = bnfinit(x^40 - x^39 - 122*x^38 + 122*x^37 + 6889*x^36 - 6889*x^35 - 238865*x^34 + 238865*x^33 + 5689120*x^32 - 5689120*x^31 - 98643416*x^30 + 98643416*x^29 + 1287488848*x^28 - 1287488848*x^27 - 12899982560*x^26 + 12899982560*x^25 + 100277346172*x^24 - 100277346172*x^23 - 607080958403*x^22 + 607080958403*x^21 + 2856692933032*x^20 - 2856692933032*x^19 - 10368625561538*x^18 + 10368625561538*x^17 + 28623261724177*x^16 - 28623261724177*x^15 - 58787122960283*x^14 + 58787122960283*x^13 + 86896851513817*x^12 - 86896851513817*x^11 - 87923917855103*x^10 + 87923917855103*x^9 + 56303216874256*x^8 - 56303216874256*x^7 - 20052325041287*x^6 + 20052325041287*x^5 + 3186318150400*x^4 - 3186318150400*x^3 - 149372259890*x^2 + 149372259890*x - 6414099449, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 122 x^{38} + 122 x^{37} + 6889 x^{36} - 6889 x^{35} - 238865 x^{34} + 238865 x^{33} + 5689120 x^{32} - 5689120 x^{31} - 98643416 x^{30} + 98643416 x^{29} + 1287488848 x^{28} - 1287488848 x^{27} - 12899982560 x^{26} + 12899982560 x^{25} + 100277346172 x^{24} - 100277346172 x^{23} - 607080958403 x^{22} + 607080958403 x^{21} + 2856692933032 x^{20} - 2856692933032 x^{19} - 10368625561538 x^{18} + 10368625561538 x^{17} + 28623261724177 x^{16} - 28623261724177 x^{15} - 58787122960283 x^{14} + 58787122960283 x^{13} + 86896851513817 x^{12} - 86896851513817 x^{11} - 87923917855103 x^{10} + 87923917855103 x^{9} + 56303216874256 x^{8} - 56303216874256 x^{7} - 20052325041287 x^{6} + 20052325041287 x^{5} + 3186318150400 x^{4} - 3186318150400 x^{3} - 149372259890 x^{2} + 149372259890 x - 6414099449 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(532628149472941385369669439648689139945981030063572028866179399432692728725190524361=11^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.93$
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:  $11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(451=11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{451}(384,·)$, $\chi_{451}(1,·)$, $\chi_{451}(133,·)$, $\chi_{451}(263,·)$, $\chi_{451}(395,·)$, $\chi_{451}(142,·)$, $\chi_{451}(144,·)$, $\chi_{451}(274,·)$, $\chi_{451}(23,·)$, $\chi_{451}(408,·)$, $\chi_{451}(153,·)$, $\chi_{451}(155,·)$, $\chi_{451}(417,·)$, $\chi_{451}(419,·)$, $\chi_{451}(166,·)$, $\chi_{451}(45,·)$, $\chi_{451}(430,·)$, $\chi_{451}(175,·)$, $\chi_{451}(54,·)$, $\chi_{451}(439,·)$, $\chi_{451}(441,·)$, $\chi_{451}(186,·)$, $\chi_{451}(320,·)$, $\chi_{451}(65,·)$, $\chi_{451}(76,·)$, $\chi_{451}(78,·)$, $\chi_{451}(208,·)$, $\chi_{451}(210,·)$, $\chi_{451}(340,·)$, $\chi_{451}(219,·)$, $\chi_{451}(221,·)$, $\chi_{451}(353,·)$, $\chi_{451}(100,·)$, $\chi_{451}(362,·)$, $\chi_{451}(364,·)$, $\chi_{451}(109,·)$, $\chi_{451}(120,·)$, $\chi_{451}(122,·)$, $\chi_{451}(252,·)$, $\chi_{451}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3085911001} a^{21} + \frac{1404325267}{3085911001} a^{20} - \frac{63}{3085911001} a^{19} - \frac{939918993}{3085911001} a^{18} + \frac{1701}{3085911001} a^{17} + \frac{823601814}{3085911001} a^{16} - \frac{25704}{3085911001} a^{15} + \frac{1079372630}{3085911001} a^{14} + \frac{238140}{3085911001} a^{13} + \frac{627943566}{3085911001} a^{12} - \frac{1393119}{3085911001} a^{11} + \frac{1251606253}{3085911001} a^{10} + \frac{5108103}{3085911001} a^{9} + \frac{1487535260}{3085911001} a^{8} - \frac{11258676}{3085911001} a^{7} + \frac{1526580906}{3085911001} a^{6} + \frac{13640319}{3085911001} a^{5} - \frac{273952974}{3085911001} a^{4} - \frac{7577955}{3085911001} a^{3} + \frac{380156627}{3085911001} a^{2} + \frac{1240029}{3085911001} a - \frac{1195455578}{3085911001}$, $\frac{1}{3085911001} a^{22} - \frac{66}{3085911001} a^{20} + \frac{1127064800}{3085911001} a^{19} + \frac{1881}{3085911001} a^{18} + \frac{561437421}{3085911001} a^{17} - \frac{30294}{3085911001} a^{16} - \frac{1130854100}{3085911001} a^{15} + \frac{302940}{3085911001} a^{14} + \frac{955860558}{3085911001} a^{13} - \frac{1945944}{3085911001} a^{12} - \frac{49525950}{3085911001} a^{11} + \frac{8027019}{3085911001} a^{10} + \frac{1115168336}{3085911001} a^{9} - \frac{20640906}{3085911001} a^{8} + \frac{709885840}{3085911001} a^{7} + \frac{30961359}{3085911001} a^{6} - \frac{1064828760}{3085911001} a^{5} - \frac{23816430}{3085911001} a^{4} - \frac{1237741937}{3085911001} a^{3} + \frac{7144929}{3085911001} a^{2} - \frac{986816013}{3085911001} a - \frac{354294}{3085911001}$, $\frac{1}{3085911001} a^{23} + \frac{1235202392}{3085911001} a^{20} - \frac{2277}{3085911001} a^{19} + \frac{245003903}{3085911001} a^{18} + \frac{81972}{3085911001} a^{17} + \frac{766378607}{3085911001} a^{16} - \frac{1393524}{3085911001} a^{15} + \frac{1218501115}{3085911001} a^{14} + \frac{13771296}{3085911001} a^{13} + \frac{1277906393}{3085911001} a^{12} - \frac{83918835}{3085911001} a^{11} + \frac{401584007}{3085911001} a^{10} + \frac{316493892}{3085911001} a^{9} + \frac{138060968}{3085911001} a^{8} - \frac{712111257}{3085911001} a^{7} + \frac{940359004}{3085911001} a^{6} + \frac{876444624}{3085911001} a^{5} - \frac{803172215}{3085911001} a^{4} - \frac{493000101}{3085911001} a^{3} - \frac{583766639}{3085911001} a^{2} + \frac{81487620}{3085911001} a + \frac{1333617878}{3085911001}$, $\frac{1}{3085911001} a^{24} - \frac{2484}{3085911001} a^{20} + \frac{914979574}{3085911001} a^{19} + \frac{94392}{3085911001} a^{18} + \frac{1192501496}{3085911001} a^{17} - \frac{1710234}{3085911001} a^{16} - \frac{77504206}{3085911001} a^{15} + \frac{18242496}{3085911001} a^{14} - \frac{783109167}{3085911001} a^{13} - \frac{122063760}{3085911001} a^{12} + \frac{114881029}{3085911001} a^{11} + \frac{517899096}{3085911001} a^{10} - \frac{867968780}{3085911001} a^{9} - \frac{1359485127}{3085911001} a^{8} + \frac{134797468}{3085911001} a^{7} - \frac{1014314617}{3085911001} a^{6} + \frac{172796569}{3085911001} a^{5} + \frac{1472456125}{3085911001} a^{4} - \frac{1125971519}{3085911001} a^{3} + \frac{488925720}{3085911001} a^{2} + \frac{300192858}{3085911001} a - \frac{24446286}{3085911001}$, $\frac{1}{3085911001} a^{25} - \frac{906399329}{3085911001} a^{20} - \frac{62100}{3085911001} a^{19} - \frac{617560360}{3085911001} a^{18} + \frac{2515050}{3085911001} a^{17} - \frac{209591893}{3085911001} a^{16} - \frac{45606240}{3085911001} a^{15} - \frac{1278156116}{3085911001} a^{14} + \frac{469476000}{3085911001} a^{13} + \frac{1541643468}{3085911001} a^{12} + \frac{143302501}{3085911001} a^{11} + \frac{609585665}{3085911001} a^{10} - \frac{1014601279}{3085911001} a^{9} + \frac{1336915111}{3085911001} a^{8} - \frac{1207666792}{3085911001} a^{7} - \frac{384853156}{3085911001} a^{6} + \frac{1409987510}{3085911001} a^{5} + \frac{361172286}{3085911001} a^{4} + \frac{180751506}{3085911001} a^{3} + \frac{320488020}{3085911001} a^{2} - \frac{30125251}{3085911001} a - \frac{865272790}{3085911001}$, $\frac{1}{3085911001} a^{26} - \frac{70200}{3085911001} a^{20} + \frac{911590932}{3085911001} a^{19} + \frac{3001050}{3085911001} a^{18} - \frac{1379833764}{3085911001} a^{17} - \frac{57999240}{3085911001} a^{16} - \frac{738451182}{3085911001} a^{15} + \frac{644436000}{3085911001} a^{14} + \frac{1261064581}{3085911001} a^{13} - \frac{1349324999}{3085911001} a^{12} - \frac{680643297}{3085911001} a^{11} + \frac{694649919}{3085911001} a^{10} - \frac{1132704364}{3085911001} a^{9} + \frac{1233511217}{3085911001} a^{8} + \frac{964834359}{3085911001} a^{7} - \frac{1197780506}{3085911001} a^{6} + \frac{446962778}{3085911001} a^{5} - \frac{460376980}{3085911001} a^{4} - \frac{1441568365}{3085911001} a^{3} + \frac{483549744}{3085911001} a^{2} + \frac{825750528}{3085911001} a - \frac{956593800}{3085911001}$, $\frac{1}{3085911001} a^{27} - \frac{1053414615}{3085911001} a^{20} - \frac{1421550}{3085911001} a^{19} - \frac{744118982}{3085911001} a^{18} + \frac{61410960}{3085911001} a^{17} - \frac{1519623118}{3085911001} a^{16} - \frac{1159984800}{3085911001} a^{15} - \frac{1324938974}{3085911001} a^{14} - \frac{61452004}{3085911001} a^{13} - \frac{1280959382}{3085911001} a^{12} - \frac{1439062850}{3085911001} a^{11} - \frac{431764236}{3085911001} a^{10} - \frac{1229245300}{3085911001} a^{9} - \frac{1288187481}{3085911001} a^{8} + \frac{1522291551}{3085911001} a^{7} - \frac{1090678750}{3085911001} a^{6} + \frac{457606510}{3085911001} a^{5} + \frac{1542926068}{3085911001} a^{4} - \frac{712199084}{3085911001} a^{3} + \frac{862629280}{3085911001} a^{2} - \frac{312066028}{3085911001} a + \frac{368096595}{3085911001}$, $\frac{1}{3085911001} a^{28} - \frac{1658475}{3085911001} a^{20} + \frac{780802295}{3085911001} a^{19} + \frac{75626460}{3085911001} a^{18} + \frac{510256417}{3085911001} a^{17} - \frac{1522480050}{3085911001} a^{16} + \frac{574830841}{3085911001} a^{15} - \frac{1115694006}{3085911001} a^{14} - \frac{1001636574}{3085911001} a^{13} + \frac{1190247790}{3085911001} a^{12} + \frac{402926138}{3085911001} a^{11} + \frac{934731726}{3085911001} a^{10} - \frac{234170851}{3085911001} a^{9} + \frac{1179317541}{3085911001} a^{8} - \frac{957684199}{3085911001} a^{7} + \frac{1328167067}{3085911001} a^{6} + \frac{966386958}{3085911001} a^{5} + \frac{593214207}{3085911001} a^{4} - \frac{531466215}{3085911001} a^{3} + \frac{54850321}{3085911001} a^{2} - \frac{1087002870}{3085911001} a - \frac{207169641}{3085911001}$, $\frac{1}{3085911001} a^{29} + \frac{260472387}{3085911001} a^{20} - \frac{28857465}{3085911001} a^{19} + \frac{870940887}{3085911001} a^{18} + \frac{1298585925}{3085911001} a^{17} + \frac{635109859}{3085911001} a^{16} - \frac{542381392}{3085911001} a^{15} - \frac{763489416}{3085911001} a^{14} + \frac{1142876162}{3085911001} a^{13} - \frac{1050158491}{3085911001} a^{12} - \frac{1256873051}{3085911001} a^{11} + \frac{895984668}{3085911001} a^{10} - \frac{1071168280}{3085911001} a^{9} + \frac{1361072849}{3085911001} a^{8} - \frac{1142955983}{3085911001} a^{7} - \frac{1415373130}{3085911001} a^{6} - \frac{92280599}{3085911001} a^{5} + \frac{158478367}{3085911001} a^{4} + \frac{1121438769}{3085911001} a^{3} - \frac{215742354}{3085911001} a^{2} + \frac{1133199468}{3085911001} a - \frac{175712071}{3085911001}$, $\frac{1}{3085911001} a^{30} - \frac{34628958}{3085911001} a^{20} - \frac{1234764738}{3085911001} a^{19} - \frac{1441035496}{3085911001} a^{18} - \frac{1143147285}{3085911001} a^{17} - \frac{115032679}{3085911001} a^{16} + \frac{1077784863}{3085911001} a^{15} - \frac{715226079}{3085911001} a^{14} - \frac{47367570}{3085911001} a^{13} - \frac{159450281}{3085911001} a^{12} + \frac{738593132}{3085911001} a^{11} + \frac{1157808097}{3085911001} a^{10} - \frac{119098853}{3085911001} a^{9} - \frac{1474434264}{3085911001} a^{8} + \frac{713446172}{3085911001} a^{7} + \frac{994434843}{3085911001} a^{6} + \frac{309176252}{3085911001} a^{5} - \frac{283095833}{3085911001} a^{4} + \frac{386294599}{3085911001} a^{3} - \frac{1161434314}{3085911001} a^{2} - \frac{442549627}{3085911001} a + \frac{384550907}{3085911001}$, $\frac{1}{3085911001} a^{31} + \frac{367827227}{3085911001} a^{20} - \frac{536748849}{3085911001} a^{19} + \frac{1450486847}{3085911001} a^{18} + \frac{156515860}{3085911001} a^{17} - \frac{255754482}{3085911001} a^{16} + \frac{1010316778}{3085911001} a^{15} - \frac{1370113353}{3085911001} a^{14} + \frac{826413167}{3085911001} a^{13} - \frac{223941192}{3085911001} a^{12} + \frac{945146728}{3085911001} a^{11} - \frac{344684544}{3085911001} a^{10} - \frac{694675911}{3085911001} a^{9} - \frac{986101322}{3085911001} a^{8} - \frac{1228038425}{3085911001} a^{7} + \frac{1310402471}{3085911001} a^{6} - \frac{1388528298}{3085911001} a^{5} - \frac{1130963294}{3085911001} a^{4} - \frac{1233063167}{3085911001} a^{3} - \frac{1012968939}{3085911001} a^{2} + \frac{845131774}{3085911001} a + \frac{898280239}{3085911001}$, $\frac{1}{3085911001} a^{32} - \frac{660613968}{3085911001} a^{20} - \frac{63685860}{3085911001} a^{19} + \frac{1416600998}{3085911001} a^{18} + \frac{510065594}{3085911001} a^{17} - \frac{262744231}{3085911001} a^{16} + \frac{1115533392}{3085911001} a^{15} + \frac{828145700}{3085911001} a^{14} - \frac{1016015587}{3085911001} a^{13} - \frac{901893601}{3085911001} a^{12} - \frac{1111393585}{3085911001} a^{11} - \frac{570345297}{3085911001} a^{10} - \frac{232108839}{3085911001} a^{9} - \frac{745983684}{3085911001} a^{8} - \frac{477414063}{3085911001} a^{7} - \frac{616831127}{3085911001} a^{6} - \frac{982665840}{3085911001} a^{5} - \frac{992182077}{3085911001} a^{4} + \frac{276159587}{3085911001} a^{3} - \frac{925685532}{3085911001} a^{2} + \frac{631224462}{3085911001} a - \frac{1004678922}{3085911001}$, $\frac{1}{3085911001} a^{33} - \frac{350272230}{3085911001} a^{20} - \frac{85235973}{3085911001} a^{19} - \frac{810218212}{3085911001} a^{18} + \frac{170010973}{3085911001} a^{17} + \frac{837749434}{3085911001} a^{16} - \frac{910960270}{3085911001} a^{15} - \frac{1087891593}{3085911001} a^{14} + \frac{1051525940}{3085911001} a^{13} + \frac{466145779}{3085911001} a^{12} - \frac{117092258}{3085911001} a^{11} + \frac{449343442}{3085911001} a^{10} - \frac{350637493}{3085911001} a^{9} + \frac{1520225678}{3085911001} a^{8} - \frac{1236295303}{3085911001} a^{7} + \frac{789072563}{3085911001} a^{6} + \frac{707833675}{3085911001} a^{5} - \frac{420497978}{3085911001} a^{4} - \frac{155743727}{3085911001} a^{3} - \frac{518132333}{3085911001} a^{2} - \frac{289057308}{3085911001} a + \frac{994734708}{3085911001}$, $\frac{1}{3085911001} a^{34} + \frac{578423812}{3085911001} a^{20} - \frac{1275991695}{3085911001} a^{19} + \frac{467266886}{3085911001} a^{18} + \frac{1069989471}{3085911001} a^{17} - \frac{776208549}{3085911001} a^{16} + \frac{203009405}{3085911001} a^{15} + \frac{56769585}{3085911001} a^{14} - \frac{965270052}{3085911001} a^{13} + \frac{1136818987}{3085911001} a^{12} - \frac{1514675800}{3085911001} a^{11} + \frac{1052079736}{3085911001} a^{10} + \frac{1521170563}{3085911001} a^{9} + \frac{647214963}{3085911001} a^{8} - \frac{913409980}{3085911001} a^{7} - \frac{1345149431}{3085911001} a^{6} + \frac{1108025122}{3085911001} a^{5} - \frac{931244191}{3085911001} a^{4} - \frac{1367311833}{3085911001} a^{3} - \frac{287594503}{3085911001} a^{2} + \frac{572616626}{3085911001} a + \frac{925685532}{3085911001}$, $\frac{1}{3085911001} a^{35} - \frac{208136473}{3085911001} a^{20} - \frac{122964970}{3085911001} a^{19} - \frac{1283103829}{3085911001} a^{18} - \frac{269503442}{3085911001} a^{17} - \frac{1511331465}{3085911001} a^{16} - \frac{56769585}{3085911001} a^{15} + \frac{546667223}{3085911001} a^{14} + \frac{1099580944}{3085911001} a^{13} + \frac{1305468484}{3085911001} a^{12} - \frac{427328763}{3085911001} a^{11} + \frac{579126468}{3085911001} a^{10} + \frac{928527791}{3085911001} a^{9} + \frac{1012477825}{3085911001} a^{8} + \frac{553925153}{3085911001} a^{7} - \frac{548437902}{3085911001} a^{6} + \frac{1278131526}{3085911001} a^{5} + \frac{185622252}{3085911001} a^{4} + \frac{1228006544}{3085911001} a^{3} + \frac{1410600141}{3085911001} a^{2} + \frac{4388415}{3085911001} a - \frac{358870111}{3085911001}$, $\frac{1}{3085911001} a^{36} - \frac{1039786676}{3085911001} a^{20} + \frac{1033853377}{3085911001} a^{19} - \frac{806599100}{3085911001} a^{18} + \frac{734954994}{3085911001} a^{17} + \frac{1295525690}{3085911001} a^{16} - \frac{1509470036}{3085911001} a^{15} - \frac{928373874}{3085911001} a^{14} + \frac{1022650642}{3085911001} a^{13} - \frac{291719165}{3085911001} a^{12} + \frac{73473143}{3085911001} a^{11} - \frac{1515601047}{3085911001} a^{10} + \frac{809266016}{3085911001} a^{9} + \frac{474853844}{3085911001} a^{8} + \frac{403279718}{3085911001} a^{7} + \frac{474598250}{3085911001} a^{6} + \frac{694224136}{3085911001} a^{5} + \frac{1209841240}{3085911001} a^{4} - \frac{1272109462}{3085911001} a^{3} + \frac{1133143435}{3085911001} a^{2} - \frac{1434783031}{3085911001} a + \frac{549456899}{3085911001}$, $\frac{1}{3085911001} a^{37} - \frac{233033508}{3085911001} a^{20} - \frac{1509028667}{3085911001} a^{19} - \frac{399606931}{3085911001} a^{18} - \frac{1340253008}{3085911001} a^{17} + \frac{1191615543}{3085911001} a^{16} - \frac{529914117}{3085911001} a^{15} - \frac{79390226}{3085911001} a^{14} + \frac{1008583235}{3085911001} a^{13} - \frac{168212580}{3085911001} a^{12} + \frac{135362916}{3085911001} a^{11} + \frac{727565793}{3085911001} a^{10} + \frac{593141315}{3085911001} a^{9} + \frac{886309792}{3085911001} a^{8} - \frac{1423539156}{3085911001} a^{7} + \frac{514609615}{3085911001} a^{6} + \frac{304907827}{3085911001} a^{5} - \frac{910851411}{3085911001} a^{4} - \frac{1450026781}{3085911001} a^{3} + \frac{1519761406}{3085911001} a^{2} - \frac{1496571321}{3085911001} a + \frac{501627754}{3085911001}$, $\frac{1}{3085911001} a^{38} + \frac{789330920}{3085911001} a^{20} + \frac{348837070}{3085911001} a^{19} - \frac{375833183}{3085911001} a^{18} - \frac{500906478}{3085911001} a^{17} + \frac{1107623854}{3085911001} a^{16} - \frac{219426917}{3085911001} a^{15} - \frac{508222878}{3085911001} a^{14} + \frac{493851557}{3085911001} a^{13} + \frac{1407014095}{3085911001} a^{12} + \frac{1329061543}{3085911001} a^{11} + \frac{420954412}{3085911001} a^{10} + \frac{738099376}{3085911001} a^{9} + \frac{672874247}{3085911001} a^{8} - \frac{544233591}{3085911001} a^{7} - \frac{1191616147}{3085911001} a^{6} - \frac{1408355412}{3085911001} a^{5} + \frac{1110181070}{3085911001} a^{4} - \frac{261121483}{3085911001} a^{3} - \frac{463794448}{3085911001} a^{2} + \frac{1017474845}{3085911001} a + \frac{1110679568}{3085911001}$, $\frac{1}{3085911001} a^{39} + \frac{482990303}{3085911001} a^{20} - \frac{22561239}{3085911001} a^{19} + \frac{707632545}{3085911001} a^{18} + \frac{827014369}{3085911001} a^{17} + \frac{1364893507}{3085911001} a^{16} - \frac{1411086773}{3085911001} a^{15} + \frac{454448696}{3085911001} a^{14} - \frac{847381793}{3085911001} a^{13} + \frac{1183043607}{3085911001} a^{12} - \frac{1203202448}{3085911001} a^{11} + \frac{1262201198}{3085911001} a^{10} + \frac{456294065}{3085911001} a^{9} + \frac{416219982}{3085911001} a^{8} + \frac{1049121969}{3085911001} a^{7} + \frac{1328677147}{3085911001} a^{6} + \frac{531840584}{3085911001} a^{5} - \frac{1091264569}{3085911001} a^{4} - \frac{142494178}{3085911001} a^{3} - \frac{644416488}{3085911001} a^{2} + \frac{216491069}{3085911001} a + \frac{161431700}{3085911001}$

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:  $39$
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, 8.8.2851397323891721.1, 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 $20^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ $20^{2}$ $40$ R $40$ $40$ $40$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
11Data not computed
41Data not computed