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Properties

Label 28.0.30734458946...5625.2
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 5^{14}\cdot 29^{26}$
Root discriminant $88.31$
Ramified primes $3, 5, 29$
Class number $178176$ (GRH)
Class group $[2, 2, 8, 8, 696]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37919821, -62913107, 277913062, -564100843, 561348952, -164626476, -229415564, 323241525, -156584897, -53198618, 146344703, -112637118, 62730879, -23680302, 4869943, 1447308, -1930463, 1057712, -392562, 101227, -5277, -13325, 9892, -4430, 1497, -402, 83, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 12*x^27 + 83*x^26 - 402*x^25 + 1497*x^24 - 4430*x^23 + 9892*x^22 - 13325*x^21 - 5277*x^20 + 101227*x^19 - 392562*x^18 + 1057712*x^17 - 1930463*x^16 + 1447308*x^15 + 4869943*x^14 - 23680302*x^13 + 62730879*x^12 - 112637118*x^11 + 146344703*x^10 - 53198618*x^9 - 156584897*x^8 + 323241525*x^7 - 229415564*x^6 - 164626476*x^5 + 561348952*x^4 - 564100843*x^3 + 277913062*x^2 - 62913107*x + 37919821)
 
gp: K = bnfinit(x^28 - 12*x^27 + 83*x^26 - 402*x^25 + 1497*x^24 - 4430*x^23 + 9892*x^22 - 13325*x^21 - 5277*x^20 + 101227*x^19 - 392562*x^18 + 1057712*x^17 - 1930463*x^16 + 1447308*x^15 + 4869943*x^14 - 23680302*x^13 + 62730879*x^12 - 112637118*x^11 + 146344703*x^10 - 53198618*x^9 - 156584897*x^8 + 323241525*x^7 - 229415564*x^6 - 164626476*x^5 + 561348952*x^4 - 564100843*x^3 + 277913062*x^2 - 62913107*x + 37919821, 1)
 

Normalized defining polynomial

\( x^{28} - 12 x^{27} + 83 x^{26} - 402 x^{25} + 1497 x^{24} - 4430 x^{23} + 9892 x^{22} - 13325 x^{21} - 5277 x^{20} + 101227 x^{19} - 392562 x^{18} + 1057712 x^{17} - 1930463 x^{16} + 1447308 x^{15} + 4869943 x^{14} - 23680302 x^{13} + 62730879 x^{12} - 112637118 x^{11} + 146344703 x^{10} - 53198618 x^{9} - 156584897 x^{8} + 323241525 x^{7} - 229415564 x^{6} - 164626476 x^{5} + 561348952 x^{4} - 564100843 x^{3} + 277913062 x^{2} - 62913107 x + 37919821 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3073445894692486733947614496733715330677488092041015625=3^{14}\cdot 5^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.31$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(386,·)$, $\chi_{435}(199,·)$, $\chi_{435}(136,·)$, $\chi_{435}(236,·)$, $\chi_{435}(266,·)$, $\chi_{435}(139,·)$, $\chi_{435}(94,·)$, $\chi_{435}(16,·)$, $\chi_{435}(209,·)$, $\chi_{435}(149,·)$, $\chi_{435}(86,·)$, $\chi_{435}(71,·)$, $\chi_{435}(349,·)$, $\chi_{435}(286,·)$, $\chi_{435}(226,·)$, $\chi_{435}(419,·)$, $\chi_{435}(296,·)$, $\chi_{435}(169,·)$, $\chi_{435}(299,·)$, $\chi_{435}(364,·)$, $\chi_{435}(49,·)$, $\chi_{435}(434,·)$, $\chi_{435}(179,·)$, $\chi_{435}(181,·)$, $\chi_{435}(254,·)$, $\chi_{435}(341,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{3}{17} a^{17} - \frac{2}{17} a^{16} - \frac{4}{17} a^{15} - \frac{4}{17} a^{14} - \frac{6}{17} a^{13} + \frac{7}{17} a^{12} - \frac{3}{17} a^{11} + \frac{1}{17} a^{10} - \frac{4}{17} a^{9} - \frac{3}{17} a^{8} - \frac{3}{17} a^{7} - \frac{7}{17} a^{6} + \frac{7}{17} a^{5} - \frac{5}{17} a^{4} + \frac{3}{17} a^{3} + \frac{8}{17} a^{2} + \frac{2}{17} a + \frac{8}{17}$, $\frac{1}{697} a^{19} + \frac{5}{697} a^{18} - \frac{200}{697} a^{17} - \frac{314}{697} a^{16} - \frac{148}{697} a^{15} - \frac{286}{697} a^{14} + \frac{250}{697} a^{13} + \frac{334}{697} a^{12} - \frac{141}{697} a^{11} - \frac{155}{697} a^{10} - \frac{79}{697} a^{9} + \frac{178}{697} a^{8} - \frac{30}{697} a^{7} + \frac{61}{697} a^{6} - \frac{297}{697} a^{5} - \frac{262}{697} a^{4} - \frac{258}{697} a^{3} + \frac{86}{697} a^{2} - \frac{209}{697} a - \frac{307}{697}$, $\frac{1}{697} a^{20} - \frac{20}{697} a^{18} - \frac{93}{697} a^{17} + \frac{315}{697} a^{16} + \frac{331}{697} a^{15} + \frac{163}{697} a^{14} - \frac{55}{697} a^{13} + \frac{321}{697} a^{12} - \frac{65}{697} a^{11} + \frac{12}{41} a^{10} - \frac{247}{697} a^{9} - \frac{141}{697} a^{8} + \frac{293}{697} a^{7} + \frac{54}{697} a^{6} - \frac{130}{697} a^{5} + \frac{27}{697} a^{4} - \frac{100}{697} a^{3} + \frac{304}{697} a^{2} - \frac{6}{17} a - \frac{310}{697}$, $\frac{1}{697} a^{21} + \frac{7}{697} a^{18} - \frac{200}{697} a^{17} + \frac{324}{697} a^{16} - \frac{9}{697} a^{15} - \frac{199}{697} a^{14} - \frac{15}{41} a^{13} + \frac{342}{697} a^{12} + \frac{172}{697} a^{11} + \frac{138}{697} a^{10} - \frac{327}{697} a^{9} - \frac{329}{697} a^{8} + \frac{151}{697} a^{7} - \frac{304}{697} a^{6} - \frac{337}{697} a^{5} + \frac{236}{697} a^{4} + \frac{23}{697} a^{3} + \frac{80}{697} a^{2} - \frac{308}{697} a + \frac{133}{697}$, $\frac{1}{697} a^{22} + \frac{11}{697} a^{18} - \frac{326}{697} a^{17} + \frac{303}{697} a^{16} - \frac{147}{697} a^{15} + \frac{66}{697} a^{14} - \frac{96}{697} a^{13} + \frac{253}{697} a^{12} - \frac{310}{697} a^{11} + \frac{307}{697} a^{10} - \frac{63}{697} a^{9} + \frac{258}{697} a^{8} - \frac{135}{697} a^{7} + \frac{302}{697} a^{6} - \frac{145}{697} a^{5} - \frac{70}{697} a^{4} - \frac{4}{17} a^{3} - \frac{336}{697} a^{2} - \frac{3}{697} a - \frac{65}{697}$, $\frac{1}{697} a^{23} - \frac{12}{697} a^{18} + \frac{125}{697} a^{17} - \frac{219}{697} a^{16} + \frac{218}{697} a^{15} + \frac{180}{697} a^{14} + \frac{168}{697} a^{13} - \frac{7}{697} a^{12} + \frac{54}{697} a^{11} - \frac{80}{697} a^{10} + \frac{348}{697} a^{9} + \frac{285}{697} a^{8} + \frac{222}{697} a^{7} + \frac{86}{697} a^{6} + \frac{12}{41} a^{5} + \frac{176}{697} a^{4} + \frac{124}{697} a^{3} - \frac{88}{697} a^{2} + \frac{184}{697} a + \frac{56}{697}$, $\frac{1}{697} a^{24} - \frac{20}{697} a^{18} + \frac{251}{697} a^{17} + \frac{345}{697} a^{16} - \frac{79}{697} a^{15} + \frac{344}{697} a^{14} + \frac{1}{17} a^{13} - \frac{161}{697} a^{12} + \frac{237}{697} a^{11} - \frac{19}{41} a^{10} + \frac{157}{697} a^{9} + \frac{185}{697} a^{8} + \frac{341}{697} a^{7} + \frac{280}{697} a^{6} + \frac{56}{697} a^{5} + \frac{96}{697} a^{4} - \frac{314}{697} a^{3} + \frac{273}{697} a^{2} - \frac{74}{697} a + \frac{252}{697}$, $\frac{1}{243253} a^{25} + \frac{106}{243253} a^{24} + \frac{157}{243253} a^{23} - \frac{75}{243253} a^{22} - \frac{142}{243253} a^{21} + \frac{8}{14309} a^{20} - \frac{101}{243253} a^{19} + \frac{6227}{243253} a^{18} + \frac{6956}{243253} a^{17} - \frac{7289}{243253} a^{16} + \frac{75536}{243253} a^{15} + \frac{96015}{243253} a^{14} + \frac{55671}{243253} a^{13} + \frac{49543}{243253} a^{12} + \frac{72855}{243253} a^{11} + \frac{228}{697} a^{10} + \frac{115191}{243253} a^{9} + \frac{109884}{243253} a^{8} - \frac{114690}{243253} a^{7} - \frac{100342}{243253} a^{6} - \frac{73954}{243253} a^{5} + \frac{1185}{14309} a^{4} - \frac{10987}{243253} a^{3} - \frac{98930}{243253} a^{2} + \frac{73002}{243253} a - \frac{91781}{243253}$, $\frac{1}{247342087367891434049} a^{26} - \frac{6309158521218}{6032733838241254489} a^{25} + \frac{40994371601275762}{247342087367891434049} a^{24} + \frac{357457509089279}{14549534551052437297} a^{23} + \frac{11063017485334734}{247342087367891434049} a^{22} - \frac{175364921384044853}{247342087367891434049} a^{21} - \frac{5572325565942511}{14549534551052437297} a^{20} + \frac{76099644130151701}{247342087367891434049} a^{19} + \frac{1335107150203603448}{247342087367891434049} a^{18} - \frac{120699537932248803856}{247342087367891434049} a^{17} + \frac{29840908195197283298}{247342087367891434049} a^{16} - \frac{41140123933234819141}{247342087367891434049} a^{15} + \frac{103945802551808613777}{247342087367891434049} a^{14} - \frac{114689705362267343603}{247342087367891434049} a^{13} + \frac{15815398076431125109}{247342087367891434049} a^{12} - \frac{94877890195851678637}{247342087367891434049} a^{11} - \frac{14892554980990624306}{247342087367891434049} a^{10} - \frac{43124580211717941824}{247342087367891434049} a^{9} + \frac{35231564221445613629}{247342087367891434049} a^{8} + \frac{123649120306199827484}{247342087367891434049} a^{7} - \frac{122054954220369011822}{247342087367891434049} a^{6} - \frac{116425859387035810546}{247342087367891434049} a^{5} + \frac{102903734876872887045}{247342087367891434049} a^{4} - \frac{28356731577580013837}{247342087367891434049} a^{3} + \frac{4593237705534423485}{14549534551052437297} a^{2} - \frac{36817959012655562154}{247342087367891434049} a + \frac{40629143012942006855}{247342087367891434049}$, $\frac{1}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{27} - \frac{1345653358926530911093035463672008541580078196087465401852713580418818}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{26} - \frac{4449739310810248080417320105356012189275537121793611876545964042547153203373907365362}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{25} + \frac{2159070344045094382410768875067879680996945606575490242163481423759454016291650623707197}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{24} + \frac{1132321795525008620597915323934975591313593996369217834299048911093417727032558087792663}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{23} - \frac{6742861291849061541379494136283662476772493704613674053297223780693025792109872215269}{15897841289460870032459686158623507974028484618965247308067326055053919341103593879038273} a^{22} - \frac{233096193249251750755585911305720683546189503225767823122102650603975611556426295168504}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{21} - \frac{963801324073838400109396730654342665405576580295279990227743285174920329544522227555949}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{20} - \frac{1348421779221334884810615488664674200862522689945148662357974447027841762820608692346854}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{19} - \frac{678667436184067082116479579679807424334832286685494909129167779358411543148567716105249}{74060675275293321370726830641392439586327818590789322825386811622324355954897230021861223} a^{18} + \frac{978159819479097891244478490754968201593921277979295882427539777457610912563543330708428429}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{17} + \frac{1501782510418030760029616983100350589574959320336984045975267263775195185616196667488365714}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{16} + \frac{1459444760792482866238905411147287995426495564277430986259881831276960628551594871295147}{3744127849922350402219235581130813838519655440471470081184783324926385442849305093583613} a^{15} + \frac{1158867833900058030644282834887479382242906733445357356141942296319807406035936294010125531}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{14} + \frac{138292800246599044793860024230959368044016275137321950672968806851264465340401822253665085}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{13} - \frac{368870631755096951273250111181933916342533247654573206370783806039165575394115240868644419}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{12} + \frac{1048376471676170811215432367771530197176009849993759868045755722045711132310651188828587586}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{11} - \frac{625666190681345710834581557214551495496818088367684169663362127545012896085632312299802439}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{10} + \frac{461030497446459274865129306128692256364054572603052368753396646869787872995137544017621360}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{9} + \frac{528639478410218276250250641940891839853674404998519096203207502842195286249402667421720198}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{8} + \frac{673298676225960353409591844705926017241520609161997760584089190993216605366784854598464748}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{7} + \frac{672523770125490577579767633065711850958707059571991186061470299311380185116900189517413648}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{6} + \frac{1166828026901952431343261336325779225804716851296484577705880077226875278041042904625435684}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{5} + \frac{512802180270178169602677606033800838699394052746110103760620583871594948066962200052051180}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{4} + \frac{18068365943608963752515647310731629373755673544318153641218487385998333403230769119106270}{74060675275293321370726830641392439586327818590789322825386811622324355954897230021861223} a^{3} - \frac{793659356014031543607587516576716643150990067048300357018022407983279805218953140201780573}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a^{2} - \frac{501345714024845641405153816642885403388754582737526265823713539373031574589152850266032702}{3036487686287026176199800056297090023039440562222362235840859276515298594150786430896310143} a - \frac{1114218256394519603339883586004284976278834233565204078006748170167628426661530376599460}{2411825008965072419539158106669650534582558031947865159524113801838998089079258483634877}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{696}$, which has order $178176$ (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:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  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:  \( 34681517373.86067 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-435}) \), \(\Q(\sqrt{5}, \sqrt{-87})\), 7.7.594823321.1, 14.14.27641779937927268828125.1, 14.0.22439994995240462987343.1, 14.0.1753124609003161170886171875.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
5Data not computed
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$