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Properties

Label 45.45.217...321.1
Degree $45$
Signature $[45, 0]$
Discriminant $2.177\times 10^{99}$
Root discriminant $161.25$
Ramified prime $181$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{45}$ (as 45T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - x^44 - 88*x^43 + 83*x^42 + 3486*x^41 - 3092*x^40 - 82393*x^39 + 68502*x^38 + 1298642*x^37 - 1008411*x^36 - 14455343*x^35 + 10441803*x^34 + 117428375*x^33 - 78557259*x^32 - 709738502*x^31 + 437535596*x^30 + 3225742725*x^29 - 1822257498*x^28 - 11074210163*x^27 + 5697019051*x^26 + 28704796942*x^25 - 13361047180*x^24 - 55904143642*x^23 + 23423035295*x^22 + 81043444290*x^21 - 30561670072*x^20 - 86189587932*x^19 + 29605929893*x^18 + 65831554861*x^17 - 21311114591*x^16 - 35027996039*x^15 + 11388975649*x^14 + 12410963786*x^13 - 4429059287*x^12 - 2711901594*x^11 + 1176564667*x^10 + 302440276*x^9 - 189664142*x^8 - 3791931*x^7 + 14878752*x^6 - 2048263*x^5 - 256726*x^4 + 86268*x^3 - 7415*x^2 + 215*x - 1)
 
gp: K = bnfinit(x^45 - x^44 - 88*x^43 + 83*x^42 + 3486*x^41 - 3092*x^40 - 82393*x^39 + 68502*x^38 + 1298642*x^37 - 1008411*x^36 - 14455343*x^35 + 10441803*x^34 + 117428375*x^33 - 78557259*x^32 - 709738502*x^31 + 437535596*x^30 + 3225742725*x^29 - 1822257498*x^28 - 11074210163*x^27 + 5697019051*x^26 + 28704796942*x^25 - 13361047180*x^24 - 55904143642*x^23 + 23423035295*x^22 + 81043444290*x^21 - 30561670072*x^20 - 86189587932*x^19 + 29605929893*x^18 + 65831554861*x^17 - 21311114591*x^16 - 35027996039*x^15 + 11388975649*x^14 + 12410963786*x^13 - 4429059287*x^12 - 2711901594*x^11 + 1176564667*x^10 + 302440276*x^9 - 189664142*x^8 - 3791931*x^7 + 14878752*x^6 - 2048263*x^5 - 256726*x^4 + 86268*x^3 - 7415*x^2 + 215*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 215, -7415, 86268, -256726, -2048263, 14878752, -3791931, -189664142, 302440276, 1176564667, -2711901594, -4429059287, 12410963786, 11388975649, -35027996039, -21311114591, 65831554861, 29605929893, -86189587932, -30561670072, 81043444290, 23423035295, -55904143642, -13361047180, 28704796942, 5697019051, -11074210163, -1822257498, 3225742725, 437535596, -709738502, -78557259, 117428375, 10441803, -14455343, -1008411, 1298642, 68502, -82393, -3092, 3486, 83, -88, -1, 1]);
 

\( x^{45} - x^{44} - 88 x^{43} + 83 x^{42} + 3486 x^{41} - 3092 x^{40} - 82393 x^{39} + 68502 x^{38} + 1298642 x^{37} - 1008411 x^{36} - 14455343 x^{35} + 10441803 x^{34} + 117428375 x^{33} - 78557259 x^{32} - 709738502 x^{31} + 437535596 x^{30} + 3225742725 x^{29} - 1822257498 x^{28} - 11074210163 x^{27} + 5697019051 x^{26} + 28704796942 x^{25} - 13361047180 x^{24} - 55904143642 x^{23} + 23423035295 x^{22} + 81043444290 x^{21} - 30561670072 x^{20} - 86189587932 x^{19} + 29605929893 x^{18} + 65831554861 x^{17} - 21311114591 x^{16} - 35027996039 x^{15} + 11388975649 x^{14} + 12410963786 x^{13} - 4429059287 x^{12} - 2711901594 x^{11} + 1176564667 x^{10} + 302440276 x^{9} - 189664142 x^{8} - 3791931 x^{7} + 14878752 x^{6} - 2048263 x^{5} - 256726 x^{4} + 86268 x^{3} - 7415 x^{2} + 215 x - 1 \)

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(217\!\cdots\!321\)\(\medspace = 181^{44}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $161.25$
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:  $181$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $45$
This field is Galois and abelian over $\Q$.
Conductor:  \(181\)
Dirichlet character group:    $\lbrace$$\chi_{181}(1,·)$, $\chi_{181}(3,·)$, $\chi_{181}(132,·)$, $\chi_{181}(5,·)$, $\chi_{181}(129,·)$, $\chi_{181}(9,·)$, $\chi_{181}(13,·)$, $\chi_{181}(14,·)$, $\chi_{181}(15,·)$, $\chi_{181}(16,·)$, $\chi_{181}(145,·)$, $\chi_{181}(148,·)$, $\chi_{181}(25,·)$, $\chi_{181}(27,·)$, $\chi_{181}(29,·)$, $\chi_{181}(161,·)$, $\chi_{181}(34,·)$, $\chi_{181}(38,·)$, $\chi_{181}(39,·)$, $\chi_{181}(169,·)$, $\chi_{181}(42,·)$, $\chi_{181}(43,·)$, $\chi_{181}(44,·)$, $\chi_{181}(45,·)$, $\chi_{181}(48,·)$, $\chi_{181}(177,·)$, $\chi_{181}(59,·)$, $\chi_{181}(62,·)$, $\chi_{181}(65,·)$, $\chi_{181}(70,·)$, $\chi_{181}(73,·)$, $\chi_{181}(75,·)$, $\chi_{181}(80,·)$, $\chi_{181}(81,·)$, $\chi_{181}(82,·)$, $\chi_{181}(87,·)$, $\chi_{181}(135,·)$, $\chi_{181}(144,·)$, $\chi_{181}(102,·)$, $\chi_{181}(125,·)$, $\chi_{181}(114,·)$, $\chi_{181}(117,·)$, $\chi_{181}(121,·)$, $\chi_{181}(170,·)$, $\chi_{181}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{19} a^{17} - \frac{4}{19} a^{16} - \frac{3}{19} a^{15} - \frac{7}{19} a^{14} + \frac{9}{19} a^{13} + \frac{2}{19} a^{12} - \frac{8}{19} a^{11} - \frac{6}{19} a^{10} + \frac{5}{19} a^{9} - \frac{1}{19} a^{8} + \frac{4}{19} a^{7} + \frac{3}{19} a^{6} + \frac{7}{19} a^{5} - \frac{9}{19} a^{4} - \frac{2}{19} a^{3} + \frac{8}{19} a^{2} + \frac{6}{19} a - \frac{5}{19}$, $\frac{1}{19} a^{18} - \frac{1}{19}$, $\frac{1}{19} a^{19} - \frac{1}{19} a$, $\frac{1}{19} a^{20} - \frac{1}{19} a^{2}$, $\frac{1}{19} a^{21} - \frac{1}{19} a^{3}$, $\frac{1}{19} a^{22} - \frac{1}{19} a^{4}$, $\frac{1}{19} a^{23} - \frac{1}{19} a^{5}$, $\frac{1}{19} a^{24} - \frac{1}{19} a^{6}$, $\frac{1}{19} a^{25} - \frac{1}{19} a^{7}$, $\frac{1}{19} a^{26} - \frac{1}{19} a^{8}$, $\frac{1}{19} a^{27} - \frac{1}{19} a^{9}$, $\frac{1}{19} a^{28} - \frac{1}{19} a^{10}$, $\frac{1}{19} a^{29} - \frac{1}{19} a^{11}$, $\frac{1}{19} a^{30} - \frac{1}{19} a^{12}$, $\frac{1}{19} a^{31} - \frac{1}{19} a^{13}$, $\frac{1}{19} a^{32} - \frac{1}{19} a^{14}$, $\frac{1}{361} a^{33} - \frac{1}{361} a^{32} + \frac{3}{361} a^{31} - \frac{7}{361} a^{30} - \frac{4}{361} a^{29} + \frac{3}{361} a^{28} + \frac{3}{361} a^{27} - \frac{8}{361} a^{26} + \frac{8}{361} a^{25} + \frac{8}{361} a^{24} + \frac{5}{361} a^{23} - \frac{5}{361} a^{22} - \frac{1}{361} a^{21} + \frac{8}{361} a^{19} - \frac{6}{361} a^{18} + \frac{3}{361} a^{17} - \frac{145}{361} a^{16} - \frac{105}{361} a^{15} + \frac{56}{361} a^{14} + \frac{119}{361} a^{13} - \frac{177}{361} a^{12} + \frac{37}{361} a^{11} + \frac{131}{361} a^{10} + \frac{69}{361} a^{9} - \frac{33}{361} a^{8} - \frac{129}{361} a^{7} + \frac{39}{361} a^{6} - \frac{98}{361} a^{5} - \frac{22}{361} a^{4} + \frac{128}{361} a^{3} - \frac{71}{361} a^{2} - \frac{47}{361} a - \frac{123}{361}$, $\frac{1}{361} a^{34} + \frac{2}{361} a^{32} - \frac{4}{361} a^{31} + \frac{8}{361} a^{30} - \frac{1}{361} a^{29} + \frac{6}{361} a^{28} - \frac{5}{361} a^{27} - \frac{3}{361} a^{25} - \frac{6}{361} a^{24} - \frac{6}{361} a^{22} - \frac{1}{361} a^{21} + \frac{8}{361} a^{20} + \frac{2}{361} a^{19} - \frac{3}{361} a^{18} - \frac{9}{361} a^{17} - \frac{60}{361} a^{16} - \frac{87}{361} a^{15} - \frac{34}{361} a^{14} + \frac{56}{361} a^{13} + \frac{107}{361} a^{12} - \frac{174}{361} a^{11} + \frac{124}{361} a^{10} - \frac{21}{361} a^{9} + \frac{66}{361} a^{8} + \frac{100}{361} a^{7} - \frac{2}{361} a^{6} + \frac{89}{361} a^{5} - \frac{8}{361} a^{4} + \frac{8}{19} a^{3} - \frac{137}{361} a^{2} - \frac{94}{361} a - \frac{66}{361}$, $\frac{1}{361} a^{35} - \frac{2}{361} a^{32} + \frac{2}{361} a^{31} - \frac{6}{361} a^{30} - \frac{5}{361} a^{29} + \frac{8}{361} a^{28} - \frac{6}{361} a^{27} - \frac{6}{361} a^{26} - \frac{3}{361} a^{25} + \frac{3}{361} a^{24} + \frac{3}{361} a^{23} + \frac{9}{361} a^{22} - \frac{9}{361} a^{21} + \frac{2}{361} a^{20} + \frac{3}{361} a^{18} - \frac{9}{361} a^{17} - \frac{25}{361} a^{16} + \frac{5}{361} a^{15} - \frac{94}{361} a^{14} + \frac{21}{361} a^{13} - \frac{48}{361} a^{12} - \frac{26}{361} a^{11} + \frac{78}{361} a^{10} - \frac{148}{361} a^{9} + \frac{128}{361} a^{8} + \frac{104}{361} a^{7} + \frac{163}{361} a^{6} - \frac{154}{361} a^{5} + \frac{44}{361} a^{4} - \frac{127}{361} a^{3} + \frac{143}{361} a^{2} - \frac{10}{361} a - \frac{39}{361}$, $\frac{1}{361} a^{36} - \frac{2}{361} a^{18} + \frac{1}{361}$, $\frac{1}{361} a^{37} - \frac{2}{361} a^{19} + \frac{1}{361} a$, $\frac{1}{361} a^{38} - \frac{2}{361} a^{20} + \frac{1}{361} a^{2}$, $\frac{1}{361} a^{39} - \frac{2}{361} a^{21} + \frac{1}{361} a^{3}$, $\frac{1}{361} a^{40} - \frac{2}{361} a^{22} + \frac{1}{361} a^{4}$, $\frac{1}{6859} a^{41} + \frac{6}{6859} a^{40} + \frac{2}{6859} a^{39} - \frac{3}{6859} a^{38} + \frac{1}{6859} a^{37} + \frac{3}{6859} a^{36} - \frac{5}{6859} a^{35} + \frac{4}{6859} a^{34} + \frac{6}{6859} a^{33} - \frac{26}{6859} a^{32} - \frac{8}{6859} a^{31} - \frac{170}{6859} a^{30} - \frac{98}{6859} a^{29} - \frac{93}{6859} a^{28} - \frac{48}{6859} a^{27} + \frac{153}{6859} a^{26} - \frac{120}{6859} a^{25} - \frac{29}{6859} a^{24} - \frac{82}{6859} a^{23} - \frac{16}{6859} a^{22} - \frac{159}{6859} a^{21} - \frac{86}{6859} a^{20} + \frac{54}{6859} a^{19} + \frac{83}{6859} a^{18} - \frac{163}{6859} a^{17} - \frac{3113}{6859} a^{16} + \frac{1733}{6859} a^{15} + \frac{955}{6859} a^{14} - \frac{1238}{6859} a^{13} + \frac{2304}{6859} a^{12} - \frac{534}{6859} a^{11} - \frac{1844}{6859} a^{10} + \frac{1279}{6859} a^{9} + \frac{528}{6859} a^{8} - \frac{2927}{6859} a^{7} + \frac{93}{361} a^{6} + \frac{2192}{6859} a^{5} + \frac{515}{6859} a^{4} + \frac{2222}{6859} a^{3} - \frac{2737}{6859} a^{2} - \frac{1025}{6859} a - \frac{2894}{6859}$, $\frac{1}{48013} a^{42} + \frac{1}{48013} a^{41} - \frac{4}{6859} a^{40} + \frac{44}{48013} a^{39} + \frac{54}{48013} a^{38} - \frac{40}{48013} a^{37} - \frac{1}{48013} a^{36} - \frac{4}{6859} a^{35} - \frac{33}{48013} a^{34} - \frac{37}{48013} a^{33} - \frac{26}{6859} a^{32} + \frac{972}{48013} a^{31} + \frac{809}{48013} a^{30} + \frac{625}{48013} a^{29} + \frac{141}{6859} a^{28} + \frac{1248}{48013} a^{27} - \frac{1056}{48013} a^{26} - \frac{132}{48013} a^{25} + \frac{158}{48013} a^{24} + \frac{97}{6859} a^{23} - \frac{212}{48013} a^{22} + \frac{1108}{48013} a^{21} + \frac{864}{48013} a^{20} + \frac{3}{48013} a^{19} - \frac{122}{48013} a^{18} + \frac{87}{6859} a^{17} - \frac{5274}{48013} a^{16} - \frac{21694}{48013} a^{15} - \frac{6887}{48013} a^{14} + \frac{904}{6859} a^{13} - \frac{2463}{6859} a^{12} + \frac{16425}{48013} a^{11} - \frac{7893}{48013} a^{10} - \frac{12688}{48013} a^{9} - \frac{510}{2527} a^{8} + \frac{2152}{48013} a^{7} - \frac{15516}{48013} a^{6} - \frac{591}{6859} a^{5} - \frac{22982}{48013} a^{4} - \frac{11339}{48013} a^{3} + \frac{3201}{6859} a^{2} + \frac{1399}{6859} a + \frac{17795}{48013}$, $\frac{1}{35673659} a^{43} + \frac{253}{35673659} a^{42} - \frac{122}{5096237} a^{41} - \frac{4170}{35673659} a^{40} + \frac{45295}{35673659} a^{39} - \frac{26556}{35673659} a^{38} - \frac{36163}{35673659} a^{37} - \frac{1528}{5096237} a^{36} - \frac{26570}{35673659} a^{35} - \frac{3887}{35673659} a^{34} - \frac{5189}{5096237} a^{33} - \frac{379023}{35673659} a^{32} + \frac{396554}{35673659} a^{31} + \frac{198760}{35673659} a^{30} - \frac{4523}{5096237} a^{29} + \frac{443592}{35673659} a^{28} + \frac{252799}{35673659} a^{27} + \frac{676845}{35673659} a^{26} - \frac{43255}{1877561} a^{25} + \frac{117544}{5096237} a^{24} + \frac{328053}{35673659} a^{23} - \frac{637824}{35673659} a^{22} - \frac{401580}{35673659} a^{21} - \frac{330355}{35673659} a^{20} + \frac{269203}{35673659} a^{19} - \frac{9487}{5096237} a^{18} - \frac{428739}{35673659} a^{17} - \frac{9704381}{35673659} a^{16} + \frac{9907619}{35673659} a^{15} - \frac{116386}{268223} a^{14} + \frac{1658407}{5096237} a^{13} + \frac{4138536}{35673659} a^{12} - \frac{4002380}{35673659} a^{11} + \frac{1907027}{35673659} a^{10} + \frac{13070454}{35673659} a^{9} + \frac{188275}{35673659} a^{8} - \frac{1112689}{35673659} a^{7} - \frac{1114404}{5096237} a^{6} + \frac{4278868}{35673659} a^{5} - \frac{14348347}{35673659} a^{4} - \frac{2276178}{5096237} a^{3} - \frac{1395548}{5096237} a^{2} + \frac{2874110}{35673659} a - \frac{1013565}{5096237}$, $\frac{1}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{44} - \frac{2465356230657902699374373627440610608793409803586815845336280145497270585788124999188}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{43} + \frac{220534833534318432333588013264004274914947876705099363540375125623573103245207314227875}{26135756453520367357499251107215324069918151781065466796922698363297461715971314521336611919} a^{42} + \frac{7456248298528908558713080008696115547947024678945935793638614554250603430451228656298860}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{41} + \frac{7375398709874264292473384275484296569421323446069723342371558439265401146541950359938977}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{40} - \frac{651805060898995835221701116102230698405873842789368641641695883446979862214378506863683570}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{39} + \frac{608322285196760979367054686707261433516026470762765671748414065624182202833755656702123755}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{38} + \frac{316183694682281352812180369841573342659081624751203087502946382885853088770743550302455478}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{37} - \frac{625300339005379198320855344416746641194646667390962446691183900357452151775180974973227545}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{36} - \frac{350107604003381681101931831286968206257284945382178750984864217532244473969185465070113269}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{35} + \frac{411736453649081973191390107360900689426586471774883922674324548067233557760083453258600350}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{34} - \frac{43487127964430863100892740459999844205895003237297235718431407093884142609093919694758213}{70939910373840997113212253005298736761206411977177695591647324128950253229064996557913660923} a^{33} + \frac{4551769317647708578941228933164396946361161880780373438042866532697276601379892366648787587}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{32} + \frac{1363139228269208817386569577545187853057014610283616858455662661442281320043842274549896779}{70939910373840997113212253005298736761206411977177695591647324128950253229064996557913660923} a^{31} - \frac{5633147140148372062757963989245803816809255410696257262983489714255604430415699631772289663}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{30} - \frac{1189821111369853103695704814767917472184826896591869017705326341487283230385666226414278867}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{29} + \frac{11210508406178533945671244812888047374429784636000881480661647849694264409503450846180194091}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{28} - \frac{6664299001070390205421200683495236792407474263369523838040258510543762136072626333286593572}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{27} - \frac{228880053429447091030979968241328455180179333861677012105253168721836525463166622660881106}{26135756453520367357499251107215324069918151781065466796922698363297461715971314521336611919} a^{26} - \frac{8628203064005422765717471606888510152546381667820330620190376930801004481090339682305840803}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{25} - \frac{8130888777901283530444568659827965051796669682211615916820320029638176903333609652493151615}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{24} + \frac{1689006130391142608407242998100479619739995417745123144580608167196598623863886711961745794}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{23} - \frac{8631670558071416496166388443513455489911929589853651975162613938366402436684932180813371978}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{22} - \frac{5741567629301913514208519297410920872201300325424522591604024308458924431025169168553881673}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{21} + \frac{4923483709409625616518833368028386088723569257636165744022683307221667498661073765310909198}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{20} - \frac{11891888568586917503938506363620561984613418232894850835959377328888598529847932007397101452}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{19} + \frac{69933697133487213814626857530941944333659874041695256164575341596728327540139388689783978}{26135756453520367357499251107215324069918151781065466796922698363297461715971314521336611919} a^{18} - \frac{260442637138419201692829679246835817454486169468797478562955201167370987766074136590161307}{70939910373840997113212253005298736761206411977177695591647324128950253229064996557913660923} a^{17} + \frac{209755573644812949586671699154857385487137579048766274246284010314437658541910876472149628034}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{16} - \frac{190031104364109248498490639948278762631280580905948073875970858406579619493225290382569584067}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{15} - \frac{83948540811820098658896124332765668320263918389260134940946799984472111605975109127882629916}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{14} + \frac{75733036067268085130292731202303009825093208943790934186415149874690787887041558118929265153}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{13} + \frac{179835565871683817584350299498605265823482707873857260824697761267875441009464744636678834953}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{12} - \frac{55534255107544061750230890110618882874456601831403758506763532165130816765435829310036181352}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{11} + \frac{65473106442036627763996239497424399673091527977025989147846857897442525954594688208966586057}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{10} + \frac{124560622182071308663777771434533353576031247686652824859473150363512548278091630354696356091}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{9} + \frac{68475323894419021841226653952445072316076621989169750848497709716020180107578902040246689240}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{8} + \frac{187645126675603367954873391118821168144115051395329179876117812040439202116221923569189210132}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{7} - \frac{211036243618894288641772018583899613563615877410598369816200757861274144091662279844047708255}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{6} + \frac{141670494431346075489582026626415679713717873100102870885062215773196467146708333043064586151}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{5} - \frac{116852120845522812129400957689231985147136681574305439992056840026970173067892910904996105571}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{4} - \frac{108954039416978927486066589166323838925905444873296255819640596664669563516383654609970059449}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a^{3} + \frac{11098573255534018100387073020479827988317322225208332761880048821672268938210961173471187059}{26135756453520367357499251107215324069918151781065466796922698363297461715971314521336611919} a^{2} - \frac{224899566518934487246824192678683214390976883616401624744013094186040383976836015731684183317}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461} a - \frac{200365853665165608454795320781878473051749157943211820477724948573937841820678449263847992421}{496579372616886979792485771037091157328444883840243869141531268902651772603454975905395626461}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $44$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2805249427577283000000000000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{45}\cdot(2\pi)^{0}\cdot 2805249427577283000000000000000000000 \cdot 1}{2\sqrt{2176994311579143098665490695095923140252535181569079054888955137870853680040351334565004592301826321}}\approx 1.05770034135362$ (assuming GRH)

Galois group

$C_{45}$ (as 45T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 45
The 45 conjugacy class representatives for $C_{45}$
Character table for $C_{45}$ is not computed

Intermediate fields

3.3.32761.1, 5.5.1073283121.1, 9.9.1151936657823500641.1, 15.15.40504199006061377874300161158921.1

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 $45$ $45$ $15^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{15}$ $45$ $45$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{5}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{45}$ $45$ $15^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{9}$ $45$ $45$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{5}$ $45$ $45$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{9}$

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
181Data not computed