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Properties

Label 36.0.28197918371...0413.1
Degree $36$
Signature $[0, 18]$
Discriminant $13^{33}\cdot 19^{24}$
Root discriminant $74.75$
Ramified primes $13, 19$
Class number Not computed
Class group Not computed
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13841287201, -11863960458, 12146435707, -10128755357, 8722143913, -7187883304, 5960098340, -4889475633, 4015581262, -3288980695, 2694284257, -2205586425, 1805546016, -454711727, 332604279, -92113055, 61510126, -18367076, 11371361, -3583588, 2072259, -663683, 352967, -101318, 42456, -441, 7426, -363, 1309, -113, 232, -28, 41, -6, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 7*x^34 - 6*x^33 + 41*x^32 - 28*x^31 + 232*x^30 - 113*x^29 + 1309*x^28 - 363*x^27 + 7426*x^26 - 441*x^25 + 42456*x^24 - 101318*x^23 + 352967*x^22 - 663683*x^21 + 2072259*x^20 - 3583588*x^19 + 11371361*x^18 - 18367076*x^17 + 61510126*x^16 - 92113055*x^15 + 332604279*x^14 - 454711727*x^13 + 1805546016*x^12 - 2205586425*x^11 + 2694284257*x^10 - 3288980695*x^9 + 4015581262*x^8 - 4889475633*x^7 + 5960098340*x^6 - 7187883304*x^5 + 8722143913*x^4 - 10128755357*x^3 + 12146435707*x^2 - 11863960458*x + 13841287201)
 
gp: K = bnfinit(x^36 - x^35 + 7*x^34 - 6*x^33 + 41*x^32 - 28*x^31 + 232*x^30 - 113*x^29 + 1309*x^28 - 363*x^27 + 7426*x^26 - 441*x^25 + 42456*x^24 - 101318*x^23 + 352967*x^22 - 663683*x^21 + 2072259*x^20 - 3583588*x^19 + 11371361*x^18 - 18367076*x^17 + 61510126*x^16 - 92113055*x^15 + 332604279*x^14 - 454711727*x^13 + 1805546016*x^12 - 2205586425*x^11 + 2694284257*x^10 - 3288980695*x^9 + 4015581262*x^8 - 4889475633*x^7 + 5960098340*x^6 - 7187883304*x^5 + 8722143913*x^4 - 10128755357*x^3 + 12146435707*x^2 - 11863960458*x + 13841287201, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 7 x^{34} - 6 x^{33} + 41 x^{32} - 28 x^{31} + 232 x^{30} - 113 x^{29} + 1309 x^{28} - 363 x^{27} + 7426 x^{26} - 441 x^{25} + 42456 x^{24} - 101318 x^{23} + 352967 x^{22} - 663683 x^{21} + 2072259 x^{20} - 3583588 x^{19} + 11371361 x^{18} - 18367076 x^{17} + 61510126 x^{16} - 92113055 x^{15} + 332604279 x^{14} - 454711727 x^{13} + 1805546016 x^{12} - 2205586425 x^{11} + 2694284257 x^{10} - 3288980695 x^{9} + 4015581262 x^{8} - 4889475633 x^{7} + 5960098340 x^{6} - 7187883304 x^{5} + 8722143913 x^{4} - 10128755357 x^{3} + 12146435707 x^{2} - 11863960458 x + 13841287201 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28197918371845586674949922450633451482314152257891038265473838050413=13^{33}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.75$
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:  $13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(134,·)$, $\chi_{247}(7,·)$, $\chi_{247}(11,·)$, $\chi_{247}(140,·)$, $\chi_{247}(144,·)$, $\chi_{247}(20,·)$, $\chi_{247}(153,·)$, $\chi_{247}(30,·)$, $\chi_{247}(159,·)$, $\chi_{247}(163,·)$, $\chi_{247}(172,·)$, $\chi_{247}(45,·)$, $\chi_{247}(49,·)$, $\chi_{247}(178,·)$, $\chi_{247}(58,·)$, $\chi_{247}(191,·)$, $\chi_{247}(64,·)$, $\chi_{247}(68,·)$, $\chi_{247}(197,·)$, $\chi_{247}(201,·)$, $\chi_{247}(77,·)$, $\chi_{247}(210,·)$, $\chi_{247}(83,·)$, $\chi_{247}(87,·)$, $\chi_{247}(216,·)$, $\chi_{247}(220,·)$, $\chi_{247}(96,·)$, $\chi_{247}(229,·)$, $\chi_{247}(102,·)$, $\chi_{247}(106,·)$, $\chi_{247}(235,·)$, $\chi_{247}(239,·)$, $\chi_{247}(115,·)$, $\chi_{247}(121,·)$, $\chi_{247}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{8}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{9}$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{10}$, $\frac{1}{3} a^{24} - \frac{1}{3} a^{11}$, $\frac{1}{58865020699233} a^{25} + \frac{1703468906932}{19621673566411} a^{24} + \frac{323327506487}{2803096223773} a^{23} + \frac{1416963040715}{19621673566411} a^{22} - \frac{6225470524855}{58865020699233} a^{21} + \frac{36821640270}{2803096223773} a^{20} - \frac{2789951246595}{19621673566411} a^{19} + \frac{8674433872642}{58865020699233} a^{18} + \frac{769177929653}{8409288671319} a^{17} + \frac{7695055116197}{58865020699233} a^{16} + \frac{6844760772079}{58865020699233} a^{15} - \frac{70066942264}{2803096223773} a^{14} - \frac{567670532886}{19621673566411} a^{13} - \frac{1291222752106}{8409288671319} a^{12} - \frac{2714107374956}{19621673566411} a^{11} + \frac{6687801296328}{19621673566411} a^{10} + \frac{245926832071}{2803096223773} a^{9} + \frac{18976357853515}{58865020699233} a^{8} - \frac{1506379440323}{19621673566411} a^{7} - \frac{1050787254343}{2803096223773} a^{6} - \frac{29187181141294}{58865020699233} a^{5} - \frac{17115939754241}{58865020699233} a^{4} + \frac{3013653878149}{8409288671319} a^{3} - \frac{14154706598971}{58865020699233} a^{2} - \frac{6108918643760}{19621673566411} a + \frac{122898195291}{2803096223773}$, $\frac{1}{1236165434683893} a^{26} + \frac{2}{412055144894631} a^{25} + \frac{94175938654}{8409288671319} a^{24} - \frac{59300746545644}{412055144894631} a^{23} - \frac{16787747484981}{137351714964877} a^{22} + \frac{74501908055}{19621673566411} a^{21} - \frac{10696881931312}{137351714964877} a^{20} - \frac{36361381045981}{412055144894631} a^{19} - \frac{1125571181239}{58865020699233} a^{18} - \frac{22868663670134}{412055144894631} a^{17} - \frac{1410521111339}{137351714964877} a^{16} - \frac{2418175790601}{19621673566411} a^{15} + \frac{2663380872458}{412055144894631} a^{14} - \frac{25396367504729}{176595062097699} a^{13} + \frac{15560036155792}{412055144894631} a^{12} - \frac{35906317816147}{412055144894631} a^{11} + \frac{23351863089140}{58865020699233} a^{10} - \frac{44209660155147}{137351714964877} a^{9} + \frac{58434810223969}{137351714964877} a^{8} - \frac{4835259620307}{19621673566411} a^{7} - \frac{49739257129952}{412055144894631} a^{6} + \frac{118220699815111}{412055144894631} a^{5} + \frac{15532289700563}{58865020699233} a^{4} - \frac{7426686284231}{137351714964877} a^{3} + \frac{42887991777782}{137351714964877} a^{2} - \frac{7238427568775}{58865020699233} a - \frac{8922944664203}{25227866013957}$, $\frac{1}{8653158042787251} a^{27} - \frac{1}{8653158042787251} a^{26} - \frac{2}{412055144894631} a^{25} - \frac{262156237307708}{2884386014262417} a^{24} + \frac{331012339106318}{2884386014262417} a^{23} + \frac{2710606650802}{412055144894631} a^{22} + \frac{44002042309446}{961462004754139} a^{21} + \frac{125333905547098}{961462004754139} a^{20} - \frac{19647963520767}{137351714964877} a^{19} - \frac{253187596734319}{2884386014262417} a^{18} + \frac{409556209606735}{2884386014262417} a^{17} - \frac{196375587647}{58865020699233} a^{16} - \frac{88938506819753}{961462004754139} a^{15} + \frac{201130642996159}{1236165434683893} a^{14} - \frac{643892320403704}{8653158042787251} a^{13} - \frac{907802676865849}{2884386014262417} a^{12} - \frac{22100603628631}{58865020699233} a^{11} - \frac{521953929690254}{2884386014262417} a^{10} + \frac{168763786615604}{2884386014262417} a^{9} + \frac{31264268936767}{137351714964877} a^{8} - \frac{458240142100661}{961462004754139} a^{7} - \frac{398777073129753}{961462004754139} a^{6} + \frac{198529173471238}{412055144894631} a^{5} - \frac{884418429173599}{2884386014262417} a^{4} - \frac{1074598830267496}{2884386014262417} a^{3} + \frac{32795606264089}{137351714964877} a^{2} + \frac{9676764850210}{25227866013957} a - \frac{5689278584152}{25227866013957}$, $\frac{1}{60572106299510757} a^{28} - \frac{1}{60572106299510757} a^{27} + \frac{1}{8653158042787251} a^{26} - \frac{2}{20190702099836919} a^{25} + \frac{506756574912822}{6730234033278973} a^{24} - \frac{399979628814688}{2884386014262417} a^{23} - \frac{512997438808116}{6730234033278973} a^{22} + \frac{2111970012655687}{20190702099836919} a^{21} + \frac{128868783013018}{961462004754139} a^{20} - \frac{269123527436564}{6730234033278973} a^{19} - \frac{1822542835855096}{20190702099836919} a^{18} + \frac{50235966905156}{412055144894631} a^{17} + \frac{1142288630185855}{20190702099836919} a^{16} - \frac{1156648263038693}{8653158042787251} a^{15} - \frac{412265269325911}{60572106299510757} a^{14} - \frac{3988198444559321}{60572106299510757} a^{13} - \frac{100542543269512}{412055144894631} a^{12} + \frac{1006133566636873}{6730234033278973} a^{11} - \frac{3316946293368605}{20190702099836919} a^{10} + \frac{19129819902269}{961462004754139} a^{9} - \frac{5904833112063919}{20190702099836919} a^{8} - \frac{2724399866850483}{6730234033278973} a^{7} - \frac{202524449826743}{961462004754139} a^{6} + \frac{8103260524511098}{20190702099836919} a^{5} + \frac{3389538059364763}{20190702099836919} a^{4} - \frac{675967305238045}{2884386014262417} a^{3} - \frac{407204378307362}{1236165434683893} a^{2} + \frac{76245292780070}{176595062097699} a - \frac{9637523289344}{25227866013957}$, $\frac{1}{424004744096575299} a^{29} - \frac{1}{424004744096575299} a^{28} + \frac{1}{60572106299510757} a^{27} - \frac{2}{141334914698858433} a^{26} + \frac{814}{141334914698858433} a^{25} + \frac{847206095568383}{6730234033278973} a^{24} + \frac{16128270314198264}{141334914698858433} a^{23} - \frac{1201192912827310}{47111638232952811} a^{22} - \frac{1520813498934320}{20190702099836919} a^{21} + \frac{7698837795130886}{141334914698858433} a^{20} - \frac{858259817946738}{47111638232952811} a^{19} + \frac{435909857342789}{2884386014262417} a^{18} + \frac{5694534944359419}{47111638232952811} a^{17} + \frac{3736927248616249}{60572106299510757} a^{16} + \frac{23223065966532599}{424004744096575299} a^{15} + \frac{68478835873417957}{424004744096575299} a^{14} - \frac{195243770678524}{2884386014262417} a^{13} + \frac{66610715911421936}{141334914698858433} a^{12} + \frac{21591806632842359}{47111638232952811} a^{11} + \frac{4366132899104071}{20190702099836919} a^{10} + \frac{23526129583380847}{47111638232952811} a^{9} - \frac{46224164726468563}{141334914698858433} a^{8} - \frac{10021600349117384}{20190702099836919} a^{7} + \frac{17098914723748014}{47111638232952811} a^{6} + \frac{5124738040743676}{141334914698858433} a^{5} - \frac{2241322639103467}{6730234033278973} a^{4} - \frac{917268547520870}{8653158042787251} a^{3} + \frac{35044529438255}{1236165434683893} a^{2} + \frac{788722147705}{176595062097699} a + \frac{253033704907}{2803096223773}$, $\frac{1}{2968033208676027093} a^{30} - \frac{1}{2968033208676027093} a^{29} + \frac{1}{424004744096575299} a^{28} - \frac{2}{989344402892009031} a^{27} + \frac{41}{2968033208676027093} a^{26} + \frac{799}{141334914698858433} a^{25} - \frac{1373739617782521}{329781467630669677} a^{24} + \frac{27099772271178371}{989344402892009031} a^{23} - \frac{7403869341603514}{141334914698858433} a^{22} - \frac{48068093528602598}{329781467630669677} a^{21} + \frac{22940174136708803}{989344402892009031} a^{20} + \frac{1391232961888795}{20190702099836919} a^{19} + \frac{49385068479056336}{989344402892009031} a^{18} - \frac{35288708324850944}{424004744096575299} a^{17} - \frac{400502299995486382}{2968033208676027093} a^{16} - \frac{44537011588070210}{2968033208676027093} a^{15} - \frac{886026426765362}{20190702099836919} a^{14} + \frac{27540923913709373}{2968033208676027093} a^{13} - \frac{322954749897738373}{989344402892009031} a^{12} - \frac{20172858837883823}{47111638232952811} a^{11} - \frac{239686081981374908}{989344402892009031} a^{10} + \frac{383944633583889754}{989344402892009031} a^{9} + \frac{23287243211888269}{47111638232952811} a^{8} + \frac{136833120184060519}{989344402892009031} a^{7} + \frac{208468477510374821}{989344402892009031} a^{6} + \frac{11196769167646816}{141334914698858433} a^{5} - \frac{21270035978388371}{60572106299510757} a^{4} + \frac{2608735534782350}{8653158042787251} a^{3} + \frac{180676642615414}{1236165434683893} a^{2} - \frac{7575423397805}{58865020699233} a + \frac{2926183219940}{25227866013957}$, $\frac{1}{20776232460732189651} a^{31} - \frac{1}{20776232460732189651} a^{30} + \frac{1}{2968033208676027093} a^{29} - \frac{2}{6925410820244063217} a^{28} + \frac{41}{20776232460732189651} a^{27} - \frac{4}{2968033208676027093} a^{26} - \frac{39139}{6925410820244063217} a^{25} - \frac{263256862549777297}{6925410820244063217} a^{24} - \frac{57030806343288025}{989344402892009031} a^{23} + \frac{1128144742518766159}{6925410820244063217} a^{22} - \frac{249765366651976270}{2308470273414687739} a^{21} + \frac{725948962027463}{47111638232952811} a^{20} + \frac{328683942354355151}{2308470273414687739} a^{19} - \frac{251619237095495771}{2968033208676027093} a^{18} + \frac{975039531850237181}{20776232460732189651} a^{17} + \frac{2238630058219251733}{20776232460732189651} a^{16} - \frac{4064228264292649}{47111638232952811} a^{15} + \frac{1273149276088039223}{20776232460732189651} a^{14} - \frac{3282097676115689563}{20776232460732189651} a^{13} - \frac{457812742175228135}{989344402892009031} a^{12} + \frac{1334653359566837410}{6925410820244063217} a^{11} - \frac{2161152936961250507}{6925410820244063217} a^{10} - \frac{103057129307209279}{989344402892009031} a^{9} - \frac{198157975411496160}{2308470273414687739} a^{8} - \frac{131411292026517625}{2308470273414687739} a^{7} + \frac{48165336965624680}{329781467630669677} a^{6} - \frac{108448900129853135}{424004744096575299} a^{5} - \frac{16136006935958797}{60572106299510757} a^{4} + \frac{756767248205578}{8653158042787251} a^{3} + \frac{78472181055518}{412055144894631} a^{2} + \frac{11845471827065}{25227866013957} a - \frac{9621170138662}{25227866013957}$, $\frac{1}{145433627225125327557} a^{32} - \frac{1}{145433627225125327557} a^{31} + \frac{1}{20776232460732189651} a^{30} - \frac{2}{48477875741708442519} a^{29} + \frac{41}{145433627225125327557} a^{28} - \frac{4}{20776232460732189651} a^{27} + \frac{232}{145433627225125327557} a^{26} - \frac{274552}{48477875741708442519} a^{25} - \frac{17358286411980412}{6925410820244063217} a^{24} + \frac{5533437516030796174}{48477875741708442519} a^{23} - \frac{6262485545335895342}{48477875741708442519} a^{22} + \frac{128448383255673548}{989344402892009031} a^{21} - \frac{5134821439327802686}{48477875741708442519} a^{20} + \frac{1742131409214891298}{20776232460732189651} a^{19} - \frac{20926195144025049061}{145433627225125327557} a^{18} - \frac{13735535894833372829}{145433627225125327557} a^{17} + \frac{149800543019872072}{989344402892009031} a^{16} - \frac{8527882492554562390}{145433627225125327557} a^{15} - \frac{3974665569460322428}{145433627225125327557} a^{14} + \frac{1428053985452060734}{20776232460732189651} a^{13} + \frac{14816812112953105243}{48477875741708442519} a^{12} + \frac{9014465094444697060}{48477875741708442519} a^{11} + \frac{64469728191036890}{6925410820244063217} a^{10} + \frac{11919560034877333274}{48477875741708442519} a^{9} - \frac{10747743445352161166}{48477875741708442519} a^{8} - \frac{1647375878205702374}{6925410820244063217} a^{7} + \frac{523304584863343879}{2968033208676027093} a^{6} + \frac{75407754380369363}{424004744096575299} a^{5} + \frac{13017934746457096}{60572106299510757} a^{4} - \frac{712373027018018}{2884386014262417} a^{3} + \frac{141828155669285}{1236165434683893} a^{2} + \frac{6083991792158}{25227866013957} a + \frac{9714489379981}{25227866013957}$, $\frac{1}{1018035390575877292899} a^{33} - \frac{1}{1018035390575877292899} a^{32} + \frac{1}{145433627225125327557} a^{31} - \frac{2}{339345130191959097633} a^{30} + \frac{41}{1018035390575877292899} a^{29} - \frac{4}{145433627225125327557} a^{28} + \frac{232}{1018035390575877292899} a^{27} - \frac{113}{1018035390575877292899} a^{26} - \frac{91484}{16159291913902814173} a^{25} + \frac{15993779493574232761}{339345130191959097633} a^{24} - \frac{53637596142772268464}{339345130191959097633} a^{23} + \frac{248198841543432093}{2308470273414687739} a^{22} - \frac{6708087771286366902}{113115043397319699211} a^{21} + \frac{19855030970449202005}{145433627225125327557} a^{20} - \frac{74377262789992453285}{1018035390575877292899} a^{19} + \frac{146333903765858725036}{1018035390575877292899} a^{18} + \frac{92866001937048426}{2308470273414687739} a^{17} - \frac{22936453980992276278}{1018035390575877292899} a^{16} - \frac{64383299300402903806}{1018035390575877292899} a^{15} - \frac{17986172399691622919}{145433627225125327557} a^{14} - \frac{81606636679562898716}{1018035390575877292899} a^{13} - \frac{17618422610959145041}{113115043397319699211} a^{12} + \frac{22535226426468407660}{48477875741708442519} a^{11} - \frac{161413110740346358094}{339345130191959097633} a^{10} - \frac{55671533785560053245}{113115043397319699211} a^{9} + \frac{3644378334138789909}{16159291913902814173} a^{8} - \frac{2007086360122713764}{20776232460732189651} a^{7} + \frac{1003007360944332038}{2968033208676027093} a^{6} - \frac{2346659603788247}{424004744096575299} a^{5} - \frac{6253997222382646}{20190702099836919} a^{4} - \frac{212611685698558}{8653158042787251} a^{3} + \frac{569481365144336}{1236165434683893} a^{2} - \frac{11175886989833}{25227866013957} a - \frac{7523601658715}{25227866013957}$, $\frac{1}{7126247734031141050293} a^{34} - \frac{1}{7126247734031141050293} a^{33} + \frac{1}{1018035390575877292899} a^{32} - \frac{2}{2375415911343713683431} a^{31} + \frac{41}{7126247734031141050293} a^{30} - \frac{4}{1018035390575877292899} a^{29} + \frac{232}{7126247734031141050293} a^{28} - \frac{113}{7126247734031141050293} a^{27} + \frac{187}{1018035390575877292899} a^{26} - \frac{121}{2375415911343713683431} a^{25} - \frac{260955004269268063739}{2375415911343713683431} a^{24} + \frac{5012992466676817837}{48477875741708442519} a^{23} - \frac{75918682973432222780}{791805303781237894477} a^{22} - \frac{120727880962114266083}{1018035390575877292899} a^{21} - \frac{471560376963808296697}{7126247734031141050293} a^{20} + \frac{119775994603633819795}{7126247734031141050293} a^{19} + \frac{4332375658807973440}{48477875741708442519} a^{18} - \frac{843709981625913570358}{7126247734031141050293} a^{17} + \frac{752465452232555516141}{7126247734031141050293} a^{16} + \frac{145529303181297344986}{1018035390575877292899} a^{15} - \frac{34466368911429627131}{7126247734031141050293} a^{14} - \frac{463124288564761687625}{7126247734031141050293} a^{13} - \frac{100685795923576233551}{339345130191959097633} a^{12} - \frac{1015396859994103217251}{2375415911343713683431} a^{11} - \frac{351945280565346085438}{2375415911343713683431} a^{10} - \frac{18122425261829334946}{113115043397319699211} a^{9} + \frac{69905396286944406595}{145433627225125327557} a^{8} - \frac{3027932010198357268}{20776232460732189651} a^{7} - \frac{1261613382762615749}{2968033208676027093} a^{6} - \frac{59081702703048020}{141334914698858433} a^{5} - \frac{29431455650489206}{60572106299510757} a^{4} - \frac{2864796654694645}{8653158042787251} a^{3} + \frac{193095311592058}{1236165434683893} a^{2} + \frac{21686928217543}{176595062097699} a + \frac{4127396771593}{25227866013957}$, $\frac{1}{49883734138217987352051} a^{35} - \frac{1}{49883734138217987352051} a^{34} + \frac{1}{7126247734031141050293} a^{33} - \frac{2}{16627911379405995784017} a^{32} + \frac{41}{49883734138217987352051} a^{31} - \frac{4}{7126247734031141050293} a^{30} + \frac{232}{49883734138217987352051} a^{29} - \frac{113}{49883734138217987352051} a^{28} + \frac{187}{7126247734031141050293} a^{27} - \frac{121}{16627911379405995784017} a^{26} - \frac{94155941}{16627911379405995784017} a^{25} - \frac{12695375904781699249}{113115043397319699211} a^{24} + \frac{2769191222976501020017}{16627911379405995784017} a^{23} + \frac{734144619205771894234}{7126247734031141050293} a^{22} + \frac{5515804261075509519005}{49883734138217987352051} a^{21} + \frac{331728531043452539095}{49883734138217987352051} a^{20} + \frac{15132910604547239839}{339345130191959097633} a^{19} + \frac{5120640396108846043238}{49883734138217987352051} a^{18} - \frac{6079224905000008515592}{49883734138217987352051} a^{17} + \frac{355871165073458173900}{7126247734031141050293} a^{16} - \frac{3121964812752336008186}{49883734138217987352051} a^{15} - \frac{7858109209756484513141}{49883734138217987352051} a^{14} + \frac{312571781992472332625}{2375415911343713683431} a^{13} - \frac{3869313366789730454887}{16627911379405995784017} a^{12} - \frac{1443670310292758578313}{5542637126468665261339} a^{11} + \frac{52715742815188270823}{2375415911343713683431} a^{10} - \frac{285376074796997164469}{1018035390575877292899} a^{9} - \frac{59597329884720757723}{145433627225125327557} a^{8} - \frac{9351788791009279082}{20776232460732189651} a^{7} + \frac{358604435877231262}{989344402892009031} a^{6} - \frac{112383098776514572}{424004744096575299} a^{5} - \frac{22875160222275619}{60572106299510757} a^{4} + \frac{2808899845550212}{8653158042787251} a^{3} - \frac{551182323081158}{1236165434683893} a^{2} + \frac{10511041227751}{25227866013957} a + \frac{1516532799914}{8409288671319}$

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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{27961}{412055144894631} a^{28} - \frac{986415632}{137351714964877} a^{15} - \frac{276034885147312}{412055144894631} a^{2} \) (order $26$)
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

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

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

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.361.1, 3.3.61009.2, 3.3.61009.1, 4.0.2197.1, \(\Q(\zeta_{13})^+\), 6.6.286315237.1, 6.6.48387275053.2, 6.6.48387275053.1, 9.9.227081481823729.1, \(\Q(\zeta_{13})\), 12.0.180102183619590473293.1, 12.0.30437269031710789986517.1, 12.0.30437269031710789986517.2, 18.18.113290500653811459555808941573877.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
19Data not computed