Properties

Label 42.0.228...203.1
Degree $42$
Signature $[0, 21]$
Discriminant $-2.282\times 10^{75}$
Root discriminant $62.27$
Ramified primes $3, 43$
Class number $179249$ (GRH)
Class group $[179249]$ (GRH)
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 + 21*x^40 - 18*x^39 + 248*x^38 - 191*x^37 + 1996*x^36 - 1375*x^35 + 12088*x^34 - 7495*x^33 + 57373*x^32 - 31825*x^31 + 219409*x^30 - 108700*x^29 + 684645*x^28 - 300153*x^27 + 1757705*x^26 - 677756*x^25 + 3715466*x^24 - 1242500*x^23 + 6455978*x^22 - 1853597*x^21 + 9148985*x^20 - 2205194*x^19 + 10469894*x^18 - 2090336*x^17 + 9505112*x^16 - 1507895*x^15 + 6709547*x^14 - 839645*x^13 + 3559478*x^12 - 314951*x^11 + 1368367*x^10 - 93808*x^9 + 355278*x^8 - 10362*x^7 + 58036*x^6 - 2497*x^5 + 4950*x^4 + 165*x^3 + 176*x^2 - 11*x + 1)
 
gp: K = bnfinit(x^42 - x^41 + 21*x^40 - 18*x^39 + 248*x^38 - 191*x^37 + 1996*x^36 - 1375*x^35 + 12088*x^34 - 7495*x^33 + 57373*x^32 - 31825*x^31 + 219409*x^30 - 108700*x^29 + 684645*x^28 - 300153*x^27 + 1757705*x^26 - 677756*x^25 + 3715466*x^24 - 1242500*x^23 + 6455978*x^22 - 1853597*x^21 + 9148985*x^20 - 2205194*x^19 + 10469894*x^18 - 2090336*x^17 + 9505112*x^16 - 1507895*x^15 + 6709547*x^14 - 839645*x^13 + 3559478*x^12 - 314951*x^11 + 1368367*x^10 - 93808*x^9 + 355278*x^8 - 10362*x^7 + 58036*x^6 - 2497*x^5 + 4950*x^4 + 165*x^3 + 176*x^2 - 11*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -11, 176, 165, 4950, -2497, 58036, -10362, 355278, -93808, 1368367, -314951, 3559478, -839645, 6709547, -1507895, 9505112, -2090336, 10469894, -2205194, 9148985, -1853597, 6455978, -1242500, 3715466, -677756, 1757705, -300153, 684645, -108700, 219409, -31825, 57373, -7495, 12088, -1375, 1996, -191, 248, -18, 21, -1, 1]);
 

\( x^{42} - x^{41} + 21 x^{40} - 18 x^{39} + 248 x^{38} - 191 x^{37} + 1996 x^{36} - 1375 x^{35} + 12088 x^{34} - 7495 x^{33} + 57373 x^{32} - 31825 x^{31} + 219409 x^{30} - 108700 x^{29} + 684645 x^{28} - 300153 x^{27} + 1757705 x^{26} - 677756 x^{25} + 3715466 x^{24} - 1242500 x^{23} + 6455978 x^{22} - 1853597 x^{21} + 9148985 x^{20} - 2205194 x^{19} + 10469894 x^{18} - 2090336 x^{17} + 9505112 x^{16} - 1507895 x^{15} + 6709547 x^{14} - 839645 x^{13} + 3559478 x^{12} - 314951 x^{11} + 1368367 x^{10} - 93808 x^{9} + 355278 x^{8} - 10362 x^{7} + 58036 x^{6} - 2497 x^{5} + 4950 x^{4} + 165 x^{3} + 176 x^{2} - 11 x + 1 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-22\!\cdots\!203\)\(\medspace = -\,3^{21}\cdot 43^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $62.27$
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:  $3, 43$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(129=3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{129}(1,·)$, $\chi_{129}(4,·)$, $\chi_{129}(10,·)$, $\chi_{129}(11,·)$, $\chi_{129}(13,·)$, $\chi_{129}(14,·)$, $\chi_{129}(16,·)$, $\chi_{129}(17,·)$, $\chi_{129}(23,·)$, $\chi_{129}(25,·)$, $\chi_{129}(31,·)$, $\chi_{129}(35,·)$, $\chi_{129}(38,·)$, $\chi_{129}(40,·)$, $\chi_{129}(41,·)$, $\chi_{129}(44,·)$, $\chi_{129}(47,·)$, $\chi_{129}(49,·)$, $\chi_{129}(52,·)$, $\chi_{129}(53,·)$, $\chi_{129}(56,·)$, $\chi_{129}(58,·)$, $\chi_{129}(59,·)$, $\chi_{129}(64,·)$, $\chi_{129}(67,·)$, $\chi_{129}(68,·)$, $\chi_{129}(74,·)$, $\chi_{129}(79,·)$, $\chi_{129}(83,·)$, $\chi_{129}(92,·)$, $\chi_{129}(95,·)$, $\chi_{129}(97,·)$, $\chi_{129}(100,·)$, $\chi_{129}(101,·)$, $\chi_{129}(103,·)$, $\chi_{129}(107,·)$, $\chi_{129}(109,·)$, $\chi_{129}(110,·)$, $\chi_{129}(121,·)$, $\chi_{129}(122,·)$, $\chi_{129}(124,·)$$\chi_{129}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $\frac{1}{7} a^{39} + \frac{3}{7} a^{38} - \frac{3}{7} a^{37} - \frac{2}{7} a^{36} - \frac{2}{7} a^{34} + \frac{1}{7} a^{33} + \frac{3}{7} a^{32} - \frac{3}{7} a^{31} - \frac{3}{7} a^{30} + \frac{1}{7} a^{29} + \frac{2}{7} a^{28} - \frac{1}{7} a^{27} + \frac{3}{7} a^{25} + \frac{2}{7} a^{24} + \frac{3}{7} a^{23} + \frac{3}{7} a^{22} - \frac{1}{7} a^{20} + \frac{2}{7} a^{19} - \frac{2}{7} a^{18} + \frac{3}{7} a^{16} + \frac{1}{7} a^{14} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{3} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{40} + \frac{2}{7} a^{38} - \frac{1}{7} a^{36} - \frac{2}{7} a^{35} + \frac{2}{7} a^{32} - \frac{1}{7} a^{31} + \frac{3}{7} a^{30} - \frac{1}{7} a^{29} + \frac{3}{7} a^{27} + \frac{3}{7} a^{26} - \frac{3}{7} a^{24} + \frac{1}{7} a^{23} - \frac{2}{7} a^{22} - \frac{1}{7} a^{21} - \frac{2}{7} a^{20} - \frac{1}{7} a^{19} - \frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{2}{7} a^{16} + \frac{1}{7} a^{15} - \frac{3}{7} a^{14} + \frac{2}{7} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{41} + \frac{111147585452516690336004128325854480465575243576599055405945215394587619708825767103897121277}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{40} - \frac{207513935396439019960761885976717108244825643708916750352287426721930450437617192347937421228}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{39} - \frac{1710714869679526733813479015933100973633449714648478822101466909998271686355473809915689213453}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{38} + \frac{273176726555171996410475955058241598386356855704005003234372885871600739398739992480799587539}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{37} - \frac{3121349090971089348495029095759861919697021672786112521774759875238141726790001933266611986743}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{36} - \frac{217947497372244428190378137171911679671094129156863503009407865365339593734763015347669201249}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{35} - \frac{2535385980801624062524502308406696810834371284087606114644871800561141982590757626613001087127}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{34} + \frac{2016861228368882029448511829477657199463150737360318947866824759005255339523772722587687543318}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{33} - \frac{448353781257388518065970597575850345585857791173073961855016646430263151609137328215965739405}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{32} + \frac{3389197316804335101839533913119328460838466812367026443269987585479097640417517066620683624137}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{31} - \frac{2663298935328437105418473543425635591831082087381129729247855326094558781429087676376779019251}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{30} + \frac{289746674552529509174902972893066370497717540951366852115273223918426120098649274889147317878}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{29} - \frac{2418065519390828994910272906524788359353823761444174440021405047795346398227566122071458628219}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{28} - \frac{3089973100919620827050182739644206115983321752153895775197244129521537681546905420711946936103}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{27} + \frac{3038401029448077911679114807565094245065382637549830879319944648148353813578804460666088756758}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{26} + \frac{3501954765023054758838892405898668552957284597930357298794199355188112752075742533555713331445}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{25} + \frac{1069576273412495777992004263685219684246520885783134022460072127732900457000208022159251170385}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{24} + \frac{1072152932219915660951049544722658889241318766283321037808086781240866701114667406901631847989}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{23} + \frac{2154213518621243807860041577539427510698374787553339082681348263596732053366775970343948496459}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{22} - \frac{1418638936925803967374758883981375771890394107585658875586205906809707302048067148612298421304}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{21} - \frac{1900089869430304309213364030419527268258717286515516793219867115532684581179145362572863116599}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{20} - \frac{1679819317602080549812437753786373156260051319263382764893690555015947384309402558140620808534}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{19} - \frac{1689944275808186962717677419038432273100500437577879396135933393263983788349919340134697205525}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{18} - \frac{180668118428539135925818779478127723920607566450163447595022297247183356155505663905172529248}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{17} - \frac{638069711450512572878712488461183375889136063572945698199801630218983365367245945198827019136}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{16} - \frac{156534626759678106129178778871119110025932353847204443572960812312033829297193723521928587}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{15} - \frac{2161120661988512479022076848219951972136556260383249281924781446518851478103213312643178948767}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{14} + \frac{2448813501053261115028190282870285405517140999208513185750456149896274919153061313111876449863}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{13} - \frac{201201361216720169194935779501195918691315760118649772631195115003198879148582319566872858739}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{12} - \frac{167414528810546346030081230861929908186045548923838334659282864051785617409026443566389153498}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{11} - \frac{3203467797705432950037418593324241714159751696589040691212799023280678971034851709641841760364}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{10} - \frac{295269578967345909976226230314508473267027185154568754626583957640837601107341104346232562482}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{9} + \frac{37959347756491249311386153777383979738475509437076716084036909791122997855687332906564875064}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{8} + \frac{144407176638651738048663177983502587057179707693757863506058517882180168740830897242519048522}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{7} + \frac{1483713318402997262016975388935006983609501200452836877488004943062003287101272881010181954380}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{6} + \frac{3045105998232723334910567443195251233887684371930101046740111543956844604933917776452150453622}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{5} + \frac{319076822845341356131171445066340556790354224078937558322507696318779351956023499723678000212}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{4} - \frac{1963623958815482163934226390163250922867794923804482701408386786154106561778732388196735822481}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{3} - \frac{210383106899177916783097666650999289733690053071491815091011483799187124413026392579086801571}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{2} - \frac{180570288086265541850095434947724873044915594298273498211506470530812811933899755725602954187}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a + \frac{1760312626958552620763679762675612149367805507264083742693663503866401892464623011612297555505}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251}$

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

Class group and class number

$C_{179249}$, which has order $179249$ (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:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{597846318319878128399333183551408849018329181957960735727361597582815770032152124649312795518}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{41} + \frac{577795988981124824640471826537711605492341248966838615487646856364007152149949840905813563508}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{40} - \frac{12523178537587947516812830502156715084357390482079938695944660334516818962634935185634908627491}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{39} + \frac{10327789905766012198757042507499584327178405376317620070835925391776762234727449171362399773411}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{38} - \frac{147662872213227124264512137988581527124844737533294717015848649903755231880842002018632627819444}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{37} + \frac{108991870768184441507852223705156087438749259315575733158340341230283701685815424399943288185689}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{36} - \frac{1186617636690075064248609661172441085124190688531330113364009509692405104351806095589232933975246}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{35} + \frac{111375102805111616905501854631029260863277836214372108534324230333680895679820156912350930044046}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{34} - \frac{7176276574924287053599812271284023196147464905146680760728541314953353872536610423679161082439124}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{33} + \frac{4221160879150901804646939813712136497379936687075009069903013512663615416720597701029972823016169}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{32} - \frac{34011511477418845403448077207522982243843768505185106158426861648707904197398150069983905316422735}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{31} + \frac{17781023674052835310520759338772705687574979925532505474480357060781963095965477562536727396195997}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{30} - \frac{129879677482630348884877884041290230672321156455466119130088411633310488526376202928171820221677435}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{29} + \frac{60179736325889069051835690647746638988871685176993040586158996964936122317887829079136553404611482}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{28} - \frac{57805297716058912675009587970845313970740150102783420211008159471640306618917933158412631381711436}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{27} + \frac{164315467651037501807610969364678582841695744738756105417698221112849118163441604031837209901423383}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{26} - \frac{1037065571124766098947732482937786314856872959848482205874011740149350317099192420766903489005127410}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{25} + \frac{366035819258047172876028846879670410850154937281081904461850797769317853456800003178014538578917612}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{24} - \frac{2187889592177705809755854375627617364023773517612498968661432487139276629163602077263084715847620402}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{23} + \frac{659369252784439098192659841567017841912557477168697501627321757118361869969171455976668428224177962}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{22} - \frac{3793196106235633165010830374725964509819479529312004832084433996086105294006468547959656676848141619}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{21} + \frac{962024569509024513988754414666089001557122869243842080863347540602980738195955280856329633989176110}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{20} - \frac{765839771990510755476089042067251560518286807422617016693079298757635162874424850978033121272501719}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{19} + \frac{1109500222460910975382825898067345583300935046258150089650260641538188676003880854101715035688698700}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{18} - \frac{6114702256386711348859284744648636257942738380067592399321399699142231327717363855689927891148129861}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{17} + \frac{1008690123449986259182745059521354430522820886884677060188385532419647924503107571476678285925801845}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{16} - \frac{5527253785388819655300637982495123128662180903789001660199537289907339815704733599169591189090071752}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{15} + \frac{680390699414122176113404773185076021875962600978208565777215128256115672581277393584285885962611535}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{14} - \frac{3879891649855407417856547542274509226366749316361384369483144455003239758556154500009228667010137815}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{13} + \frac{344292991904484865112204774986533908013760816670585986735283535579499305536011253394908546356274047}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{12} - \frac{2041502705469106088064360544147765855494142904361565955059986714144877106013119449820945120753504619}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{11} + \frac{14735190748452567555255661542726614853539489295614533436726856395202867091834533097114007606974213}{1039303173611034910827325344230253334624903012305619963629698492955647557570270536914668133893} a^{10} - \frac{776039479458341761024115479694784652012709720856973163900366310208593094183999524083856481030081622}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{9} + \frac{22843770188397564722382340739256632224553285769652651612711625117454068806568482849212928259510045}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{8} - \frac{197471240484244062769587273428737177691653362276355366307396018300114334383824412818805983219246281}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{7} - \frac{2894967940651568123738247656452587131682002710003036109813463424300992072517509281661900053153640}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{6} - \frac{31360340777808760321179670174132844208734594973190295847305277271760256007499131289132288512760277}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{5} - \frac{19624106787617821707454577878869029363926440443349242484213444581441251520676985646440768640197}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{4} - \frac{2464881776253411541764160269476787438853051259322567633434722858421402161395445445634484092633078}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{3} - \frac{286967956216437266294491942009933809590950587838751496633549584192917925515681860216927387656964}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a^{2} - \frac{79508453895139319054498562092581723131050820465764982371314038394671956618707060343264250792079}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} a + \frac{4849027870496966572840092730071372039449466830505132381981584709487192027667730038108879054017}{7275122215277244375791277409611773342374321086139339745407889450689532902991893758402676937251} \) (order $6$)
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:  \( 2748021948787771.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{21}\cdot 2748021948787771.5 \cdot 179249}{6\sqrt{2281836760183646137444154412268560109828024514076489472840222217265158917203}}\approx 0.0993026041253754$ (assuming GRH)

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1849.1, 6.0.92307627.1, 7.7.6321363049.1, 14.0.87391712553613254588987.1, \(\Q(\zeta_{43})^+\)

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{3}$ R $42$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{3}$ $21^{2}$ $42$ $21^{2}$ $42$ $42$ $21^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{3}$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
43Data not computed