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Properties

Label 42.0.73030361130...4827.1
Degree $42$
Signature $[0, 21]$
Discriminant $-\,3^{63}\cdot 43^{36}$
Root discriminant $130.56$
Ramified primes $3, 43$
Class number $52315739$ (GRH)
Class group $[52315739]$ (GRH)
Galois group $C_{42}$ (as 42T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13841287201, 0, 0, -44832857377, 0, 0, 177930112269, 0, 0, 110795591088, 0, 0, 67829457438, 0, 0, 16711764404, 0, 0, 5181127332, 0, 0, 899104925, 0, 0, 257988274, 0, 0, 32949172, 0, 0, 3992238, 0, 0, 136792, 0, 0, 4672, 0, 0, -50, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 50*x^39 + 4672*x^36 + 136792*x^33 + 3992238*x^30 + 32949172*x^27 + 257988274*x^24 + 899104925*x^21 + 5181127332*x^18 + 16711764404*x^15 + 67829457438*x^12 + 110795591088*x^9 + 177930112269*x^6 - 44832857377*x^3 + 13841287201)
 
gp: K = bnfinit(x^42 - 50*x^39 + 4672*x^36 + 136792*x^33 + 3992238*x^30 + 32949172*x^27 + 257988274*x^24 + 899104925*x^21 + 5181127332*x^18 + 16711764404*x^15 + 67829457438*x^12 + 110795591088*x^9 + 177930112269*x^6 - 44832857377*x^3 + 13841287201, 1)
 

Normalized defining polynomial

\( x^{42} - 50 x^{39} + 4672 x^{36} + 136792 x^{33} + 3992238 x^{30} + 32949172 x^{27} + 257988274 x^{24} + 899104925 x^{21} + 5181127332 x^{18} + 16711764404 x^{15} + 67829457438 x^{12} + 110795591088 x^{9} + 177930112269 x^{6} - 44832857377 x^{3} + 13841287201 \)

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

Invariants

Degree:  $42$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 21]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-73030361130238276622262272470733471748322455720112810864804659294987449502046181947024827=-\,3^{63}\cdot 43^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.56$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(387=3^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{387}(256,·)$, $\chi_{387}(1,·)$, $\chi_{387}(130,·)$, $\chi_{387}(259,·)$, $\chi_{387}(4,·)$, $\chi_{387}(133,·)$, $\chi_{387}(262,·)$, $\chi_{387}(385,·)$, $\chi_{387}(11,·)$, $\chi_{387}(140,·)$, $\chi_{387}(269,·)$, $\chi_{387}(16,·)$, $\chi_{387}(145,·)$, $\chi_{387}(274,·)$, $\chi_{387}(35,·)$, $\chi_{387}(164,·)$, $\chi_{387}(293,·)$, $\chi_{387}(41,·)$, $\chi_{387}(170,·)$, $\chi_{387}(299,·)$, $\chi_{387}(44,·)$, $\chi_{387}(173,·)$, $\chi_{387}(302,·)$, $\chi_{387}(47,·)$, $\chi_{387}(176,·)$, $\chi_{387}(305,·)$, $\chi_{387}(59,·)$, $\chi_{387}(188,·)$, $\chi_{387}(317,·)$, $\chi_{387}(64,·)$, $\chi_{387}(193,·)$, $\chi_{387}(322,·)$, $\chi_{387}(97,·)$, $\chi_{387}(226,·)$, $\chi_{387}(355,·)$, $\chi_{387}(107,·)$, $\chi_{387}(236,·)$, $\chi_{387}(365,·)$, $\chi_{387}(121,·)$, $\chi_{387}(250,·)$, $\chi_{387}(379,·)$, $\chi_{387}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{17} + \frac{1}{7} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{7} a^{18} + \frac{1}{7} a^{12} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{19} - \frac{1}{7} a$, $\frac{1}{7} a^{20} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{21} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{22} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{23} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{24} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{25} - \frac{1}{7} a^{7}$, $\frac{1}{49} a^{26} + \frac{2}{49} a^{20} + \frac{3}{49} a^{14} + \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{343} a^{27} - \frac{1}{49} a^{24} + \frac{23}{343} a^{21} + \frac{17}{343} a^{15} + \frac{114}{343} a^{9} - \frac{20}{49} a^{6} + \frac{92}{343} a^{3}$, $\frac{1}{343} a^{28} - \frac{1}{49} a^{25} + \frac{23}{343} a^{22} + \frac{17}{343} a^{16} + \frac{114}{343} a^{10} - \frac{20}{49} a^{7} + \frac{92}{343} a^{4}$, $\frac{1}{343} a^{29} + \frac{23}{343} a^{23} + \frac{2}{49} a^{20} + \frac{17}{343} a^{17} + \frac{3}{49} a^{14} + \frac{114}{343} a^{11} - \frac{18}{49} a^{8} + \frac{92}{343} a^{5} + \frac{1}{49} a^{2}$, $\frac{1}{12691} a^{30} - \frac{3}{12691} a^{27} - \frac{495}{12691} a^{24} - \frac{24}{343} a^{21} + \frac{507}{12691} a^{18} - \frac{226}{12691} a^{15} - \frac{2140}{12691} a^{12} - \frac{157}{343} a^{9} + \frac{3599}{12691} a^{6} + \frac{4484}{12691} a^{3} - \frac{14}{37}$, $\frac{1}{12691} a^{31} - \frac{3}{12691} a^{28} - \frac{495}{12691} a^{25} - \frac{24}{343} a^{22} + \frac{507}{12691} a^{19} - \frac{226}{12691} a^{16} - \frac{327}{12691} a^{13} - \frac{157}{343} a^{10} + \frac{5412}{12691} a^{7} + \frac{4484}{12691} a^{4} - \frac{61}{259} a$, $\frac{1}{12691} a^{32} - \frac{3}{12691} a^{29} + \frac{23}{12691} a^{26} - \frac{24}{343} a^{23} - \frac{270}{12691} a^{20} - \frac{226}{12691} a^{17} - \frac{586}{12691} a^{14} - \frac{157}{343} a^{11} + \frac{4635}{12691} a^{8} + \frac{4484}{12691} a^{5} - \frac{353}{1813} a^{2}$, $\frac{1}{12691} a^{33} + \frac{2}{1813} a^{27} - \frac{80}{1813} a^{24} + \frac{692}{12691} a^{21} - \frac{2}{49} a^{18} + \frac{549}{12691} a^{15} - \frac{193}{1813} a^{12} + \frac{1712}{12691} a^{9} - \frac{4}{49} a^{6} - \frac{3523}{12691} a^{3} - \frac{5}{37}$, $\frac{1}{88837} a^{34} - \frac{1}{88837} a^{31} + \frac{17}{88837} a^{28} + \frac{1748}{88837} a^{25} - \frac{2046}{88837} a^{22} + \frac{6227}{88837} a^{19} - \frac{2851}{88837} a^{16} - \frac{4650}{88837} a^{13} - \frac{8796}{88837} a^{10} + \frac{804}{88837} a^{7} + \frac{4684}{88837} a^{4} + \frac{83}{259} a$, $\frac{1}{621859} a^{35} - \frac{1}{621859} a^{32} + \frac{17}{621859} a^{29} + \frac{1748}{621859} a^{26} + \frac{10645}{621859} a^{23} + \frac{18918}{621859} a^{20} - \frac{2851}{621859} a^{17} - \frac{30032}{621859} a^{14} - \frac{8796}{621859} a^{11} - \frac{291089}{621859} a^{8} - \frac{274518}{621859} a^{5} + \frac{107}{259} a^{2}$, $\frac{1}{663990897962341957} a^{36} + \frac{13323056733979}{663990897962341957} a^{33} + \frac{14004887163473}{663990897962341957} a^{30} + \frac{151144753053866}{663990897962341957} a^{27} + \frac{5960126954330533}{663990897962341957} a^{24} + \frac{27184796469777762}{663990897962341957} a^{21} - \frac{26584135496821163}{663990897962341957} a^{18} + \frac{21460955249886101}{663990897962341957} a^{15} - \frac{236642469894150219}{663990897962341957} a^{12} - \frac{140889948198257763}{663990897962341957} a^{9} + \frac{92880564196419161}{663990897962341957} a^{6} - \frac{158492278116689}{1935833521756099} a^{3} + \frac{2721075253695}{5643829509493}$, $\frac{1}{4647936285736393699} a^{37} + \frac{13323056733979}{4647936285736393699} a^{34} + \frac{66324712075800}{4647936285736393699} a^{31} - \frac{3877481765195313}{4647936285736393699} a^{28} + \frac{102019325493362905}{4647936285736393699} a^{25} - \frac{108323550053149168}{4647936285736393699} a^{22} - \frac{189769669398369076}{4647936285736393699} a^{19} + \frac{133530020212090535}{4647936285736393699} a^{16} + \frac{125672317623714256}{4647936285736393699} a^{13} - \frac{32483270979916219}{4647936285736393699} a^{10} - \frac{789336323475238713}{4647936285736393699} a^{7} - \frac{1895724015511710}{13550834652292693} a^{4} + \frac{17517060724528}{39506806566451} a$, $\frac{1}{32535554000154755893} a^{38} + \frac{13323056733979}{32535554000154755893} a^{35} - \frac{666152836696778}{32535554000154755893} a^{32} - \frac{42332553075755658}{32535554000154755893} a^{29} + \frac{85172341871593611}{32535554000154755893} a^{26} - \frac{1720872873675979635}{32535554000154755893} a^{23} - \frac{1889117582550750036}{32535554000154755893} a^{20} - \frac{392022621032234180}{32535554000154755893} a^{17} - \frac{1626780217814678609}{32535554000154755893} a^{14} + \frac{4236029644492282076}{32535554000154755893} a^{11} - \frac{5038072345130577402}{32535554000154755893} a^{8} - \frac{32054365947117345}{94855842566048851} a^{5} + \frac{109496228135995}{276547645965157} a^{2}$, $\frac{1}{174828558393290412081447598278298847616673677317494109321441953} a^{39} + \frac{67875935537504094480193902760952469902560614}{174828558393290412081447598278298847616673677317494109321441953} a^{36} + \frac{3845910816707717448464896370793369110331430058750256394273}{174828558393290412081447598278298847616673677317494109321441953} a^{33} + \frac{4041785697488202766440717952453926372781991585080920092518}{174828558393290412081447598278298847616673677317494109321441953} a^{30} - \frac{135643171027642168423135345548712389142265471485488999746702}{174828558393290412081447598278298847616673677317494109321441953} a^{27} + \frac{6460868973267010815367385733194550955280999724242083364800673}{174828558393290412081447598278298847616673677317494109321441953} a^{24} - \frac{8427056406877214529497742728306176247240355995356268657879699}{174828558393290412081447598278298847616673677317494109321441953} a^{21} - \frac{12276966748226971850746922620506368541980314245236917491486966}{174828558393290412081447598278298847616673677317494109321441953} a^{18} + \frac{4550087025389343199638070026126430550525223069056145036017800}{174828558393290412081447598278298847616673677317494109321441953} a^{15} - \frac{79184005901460459101818035163172747921325734874540809948605974}{174828558393290412081447598278298847616673677317494109321441953} a^{12} + \frac{9015425297924603761899544632440042903979542768769546258293284}{174828558393290412081447598278298847616673677317494109321441953} a^{9} - \frac{23588998697690938377413534303472961353429766680529662065505}{509704251875482250966319528508159905587969904715726266243271} a^{6} - \frac{317509455917775557731694941976117067167791056796442029646}{1486018227042222306024255185154985147486792725118735470097} a^{3} + \frac{741716325771760298925478714524401589196002751790576390}{4332414656099773486951181297827945036404643513465701079}$, $\frac{1}{1223799908753032884570133187948091933316715741222458765250093671} a^{40} + \frac{67875935537504094480193902760952469902560614}{1223799908753032884570133187948091933316715741222458765250093671} a^{37} + \frac{3845910816707717448464896370793369110331430058750256394273}{1223799908753032884570133187948091933316715741222458765250093671} a^{34} + \frac{17817576288717452792557461966187977875159556577397846207201}{1223799908753032884570133187948091933316715741222458765250093671} a^{31} - \frac{1706083298427776671400444163114394260413307880609618576820564}{1223799908753032884570133187948091933316715741222458765250093671} a^{28} + \frac{60296658603790919917431621338867224226572523714216630620981837}{1223799908753032884570133187948091933316715741222458765250093671} a^{25} + \frac{19096973194398827022683511811134458654510018859292949719256935}{1223799908753032884570133187948091933316715741222458765250093671} a^{22} - \frac{80219165944169632979554704096242710551706464787343997089103522}{1223799908753032884570133187948091933316715741222458765250093671} a^{19} + \frac{75343874873716459083852017512705721221243529564572828339373737}{1223799908753032884570133187948091933316715741222458765250093671} a^{16} - \frac{8762164399096532968309239764964276641171622633816683650346478}{1223799908753032884570133187948091933316715741222458765250093671} a^{13} - \frac{320253521413636930362342870783831256105849015677589621734859782}{1223799908753032884570133187948091933316715741222458765250093671} a^{10} - \frac{638398918145321723799556733610297627237965524413437839801853}{3567929763128375756764236699557119339115789333010083863702897} a^{7} - \frac{2261943990490554977307676472563437479993075115839994249048}{10402127589295556142169786296084896032407549075831148290679} a^{4} + \frac{12099668262033328629581278333154419657608176287092549489}{30326902592698414408658269084795615254832504594259907553} a$, $\frac{1}{8566599361271230191990932315636643533217010188557211356750655697} a^{41} + \frac{67875935537504094480193902760952469902560614}{8566599361271230191990932315636643533217010188557211356750655697} a^{38} + \frac{3845910816707717448464896370793369110331430058750256394273}{8566599361271230191990932315636643533217010188557211356750655697} a^{35} + \frac{307109178704531703341009086254603059425088421416053294615544}{8566599361271230191990932315636643533217010188557211356750655697} a^{32} + \frac{993971657453156333718437663577479834052694857884498941657304}{8566599361271230191990932315636643533217010188557211356750655697} a^{29} + \frac{66950365459354647680046008697500771102220887605505705934373726}{8566599361271230191990932315636643533217010188557211356750655697} a^{26} + \frac{193925531587689239104131110089433306271183696176787059040698888}{8566599361271230191990932315636643533217010188557211356750655697} a^{23} + \frac{241280234873938604129957867696282583410781147003348424575368332}{8566599361271230191990932315636643533217010188557211356750655697} a^{20} - \frac{453866896478946409919448820518901601294093007180274205282288391}{8566599361271230191990932315636643533217010188557211356750655697} a^{17} + \frac{595953715184093855428137472205919382158696347833919422005893173}{8566599361271230191990932315636643533217010188557211356750655697} a^{14} + \frac{1552909604228760341938881396483656396929940384152704406709161143}{8566599361271230191990932315636643533217010188557211356750655697} a^{11} - \frac{6704807272789092276116844536229754021489049144907613996998379}{24975508341898630297349656896899835373810525331070587045920279} a^{8} - \frac{27082464647574159980631722538125081294772477929985359668506}{72814893125068892995188504072594272226852843530818038034753} a^{5} + \frac{64322828711236003634452274517783162528864149449408838171}{212288318148888900860607883593569306783827532159819352871} a^{2}$

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

Class group and class number

$C_{52315739}$, which has order $52315739$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $20$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{342193938936787437936549671970347818981474306697798043}{8566599361271230191990932315636643533217010188557211356750655697} a^{41} - \frac{17000356765400901854924398229160279237606230324669767863}{8566599361271230191990932315636643533217010188557211356750655697} a^{38} + \frac{1593205138320429062769788310559873431512268131570157555646}{8566599361271230191990932315636643533217010188557211356750655697} a^{35} + \frac{47323160531064388590541850776052165041055897850577605493318}{8566599361271230191990932315636643533217010188557211356750655697} a^{32} + \frac{1380804125334639056152071663326816742118392712817079879283977}{8566599361271230191990932315636643533217010188557211356750655697} a^{29} + \frac{11703752280671749495707486765846731260651979917478176195769028}{8566599361271230191990932315636643533217010188557211356750655697} a^{26} + \frac{91657643113790073076650717341753189258003583085390441273906862}{8566599361271230191990932315636643533217010188557211356750655697} a^{23} + \frac{334097380127936667443167091200809092192026995592996721211655985}{8566599361271230191990932315636643533217010188557211356750655697} a^{20} + \frac{1857323055998308735068781847003971317322088012651146174610307029}{8566599361271230191990932315636643533217010188557211356750655697} a^{17} + \frac{6243656621951391130089245110158567940205912880576922508541945459}{8566599361271230191990932315636643533217010188557211356750655697} a^{14} + \frac{24784633890781083816087655292225592094287849906108790017715311933}{8566599361271230191990932315636643533217010188557211356750655697} a^{11} + \frac{130397349640898101425881333133965513422342404799847963528988817}{24975508341898630297349656896899835373810525331070587045920279} a^{8} + \frac{594377438992624042506611331335050929660387664937041397818822}{72814893125068892995188504072594272226852843530818038034753} a^{5} + \frac{84974539764778817492643988787099469935369944058166722696}{212288318148888900860607883593569306783827532159819352871} a^{2} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 545753294074079900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 7.7.6321363049.1, 14.0.87391712553613254588987.1, 21.21.5778662528422377251527626979988196021835689.1

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 $42$ R $42$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{14}$ $42$ $21^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{6}$ $42$ $42$ $21^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{42}$ $42$ R $42$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{3}$ $42$

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
3Data not computed
43Data not computed