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Properties

Label 28.0.99041240103...0029.1
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 7^{14}\cdot 29^{27}$
Root discriminant $117.84$
Ramified primes $3, 7, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8478772712071, -8301770758946, 8301770758946, -7062757087071, 7062757087071, -4485608649571, 4485608649571, -1982093024571, 1982093024571, -605159430821, 605159430821, -129491462071, 129491462071, -19721930821, 19721930821, -2158805821, 2158805821, -170040196, 170040196, -9543321, 9543321, -372071, 372071, -9571, 9571, -146, 146, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 146*x^26 - 146*x^25 + 9571*x^24 - 9571*x^23 + 372071*x^22 - 372071*x^21 + 9543321*x^20 - 9543321*x^19 + 170040196*x^18 - 170040196*x^17 + 2158805821*x^16 - 2158805821*x^15 + 19721930821*x^14 - 19721930821*x^13 + 129491462071*x^12 - 129491462071*x^11 + 605159430821*x^10 - 605159430821*x^9 + 1982093024571*x^8 - 1982093024571*x^7 + 4485608649571*x^6 - 4485608649571*x^5 + 7062757087071*x^4 - 7062757087071*x^3 + 8301770758946*x^2 - 8301770758946*x + 8478772712071)
 
gp: K = bnfinit(x^28 - x^27 + 146*x^26 - 146*x^25 + 9571*x^24 - 9571*x^23 + 372071*x^22 - 372071*x^21 + 9543321*x^20 - 9543321*x^19 + 170040196*x^18 - 170040196*x^17 + 2158805821*x^16 - 2158805821*x^15 + 19721930821*x^14 - 19721930821*x^13 + 129491462071*x^12 - 129491462071*x^11 + 605159430821*x^10 - 605159430821*x^9 + 1982093024571*x^8 - 1982093024571*x^7 + 4485608649571*x^6 - 4485608649571*x^5 + 7062757087071*x^4 - 7062757087071*x^3 + 8301770758946*x^2 - 8301770758946*x + 8478772712071, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 146 x^{26} - 146 x^{25} + 9571 x^{24} - 9571 x^{23} + 372071 x^{22} - 372071 x^{21} + 9543321 x^{20} - 9543321 x^{19} + 170040196 x^{18} - 170040196 x^{17} + 2158805821 x^{16} - 2158805821 x^{15} + 19721930821 x^{14} - 19721930821 x^{13} + 129491462071 x^{12} - 129491462071 x^{11} + 605159430821 x^{10} - 605159430821 x^{9} + 1982093024571 x^{8} - 1982093024571 x^{7} + 4485608649571 x^{6} - 4485608649571 x^{5} + 7062757087071 x^{4} - 7062757087071 x^{3} + 8301770758946 x^{2} - 8301770758946 x + 8478772712071 \)

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:  \(9904124010343790995983966505676304133800374729508003670029=3^{14}\cdot 7^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.84$
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:  $3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(609=3\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{609}(64,·)$, $\chi_{609}(1,·)$, $\chi_{609}(524,·)$, $\chi_{609}(589,·)$, $\chi_{609}(398,·)$, $\chi_{609}(461,·)$, $\chi_{609}(272,·)$, $\chi_{609}(274,·)$, $\chi_{609}(169,·)$, $\chi_{609}(526,·)$, $\chi_{609}(22,·)$, $\chi_{609}(356,·)$, $\chi_{609}(463,·)$, $\chi_{609}(400,·)$, $\chi_{609}(482,·)$, $\chi_{609}(547,·)$, $\chi_{609}(484,·)$, $\chi_{609}(293,·)$, $\chi_{609}(230,·)$, $\chi_{609}(295,·)$, $\chi_{609}(104,·)$, $\chi_{609}(41,·)$, $\chi_{609}(566,·)$, $\chi_{609}(503,·)$, $\chi_{609}(442,·)$, $\chi_{609}(251,·)$, $\chi_{609}(188,·)$, $\chi_{609}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1850229480761} a^{15} + \frac{36018836583}{1850229480761} a^{14} + \frac{75}{1850229480761} a^{13} + \frac{671089080049}{1850229480761} a^{12} + \frac{2250}{1850229480761} a^{11} + \frac{877769634118}{1850229480761} a^{10} + \frac{34375}{1850229480761} a^{9} + \frac{27195634879}{1850229480761} a^{8} + \frac{281250}{1850229480761} a^{7} + \frac{190369444153}{1850229480761} a^{6} + \frac{1181250}{1850229480761} a^{5} - \frac{598921506664}{1850229480761} a^{4} + \frac{2187500}{1850229480761} a^{3} - \frac{748651883330}{1850229480761} a^{2} + \frac{1171875}{1850229480761} a - \frac{454864381212}{1850229480761}$, $\frac{1}{1850229480761} a^{16} + \frac{80}{1850229480761} a^{14} - \frac{180094182915}{1850229480761} a^{13} + \frac{2600}{1850229480761} a^{12} - \frac{604745004909}{1850229480761} a^{11} + \frac{44000}{1850229480761} a^{10} - \frac{316789276637}{1850229480761} a^{9} + \frac{412500}{1850229480761} a^{8} - \frac{101012358122}{1850229480761} a^{7} + \frac{2100000}{1850229480761} a^{6} + \frac{27504404542}{1850229480761} a^{5} + \frac{5250000}{1850229480761} a^{4} + \frac{68761011355}{1850229480761} a^{3} + \frac{5000000}{1850229480761} a^{2} - \frac{743840483644}{1850229480761} a + \frac{781250}{1850229480761}$, $\frac{1}{1850229480761} a^{17} + \frac{638857851967}{1850229480761} a^{14} - \frac{3400}{1850229480761} a^{13} - \frac{635216466760}{1850229480761} a^{12} - \frac{136000}{1850229480761} a^{11} - \frac{229639737159}{1850229480761} a^{10} - \frac{2337500}{1850229480761} a^{9} - \frac{426433667681}{1850229480761} a^{8} - \frac{20400000}{1850229480761} a^{7} - \frac{400215281610}{1850229480761} a^{6} - \frac{89250000}{1850229480761} a^{5} - \frac{123484955311}{1850229480761} a^{4} - \frac{170000000}{1850229480761} a^{3} - \frac{59033201596}{1850229480761} a^{2} - \frac{92968750}{1850229480761} a - \frac{615439118260}{1850229480761}$, $\frac{1}{1850229480761} a^{18} - \frac{3825}{1850229480761} a^{14} - \frac{443588864499}{1850229480761} a^{13} - \frac{165750}{1850229480761} a^{12} - \frac{31500111612}{1850229480761} a^{11} - \frac{3155625}{1850229480761} a^{10} - \frac{791387880997}{1850229480761} a^{9} - \frac{31556250}{1850229480761} a^{8} + \frac{314254661872}{1850229480761} a^{7} - \frac{167343750}{1850229480761} a^{6} + \frac{286967534248}{1850229480761} a^{5} - \frac{430312500}{1850229480761} a^{4} + \frac{769591019097}{1850229480761} a^{3} - \frac{418359375}{1850229480761} a^{2} - \frac{105453661433}{1850229480761} a - \frac{66406250}{1850229480761}$, $\frac{1}{1850229480761} a^{19} + \frac{411479489162}{1850229480761} a^{14} + \frac{121125}{1850229480761} a^{13} + \frac{615941260306}{1850229480761} a^{12} + \frac{5450625}{1850229480761} a^{11} + \frac{361184519899}{1850229480761} a^{10} + \frac{99928125}{1850229480761} a^{9} + \frac{724707151431}{1850229480761} a^{8} + \frac{908437500}{1850229480761} a^{7} - \frac{540324000361}{1850229480761} a^{6} + \frac{4087968750}{1850229480761} a^{5} + \frac{478925211415}{1850229480761} a^{4} + \frac{7948828125}{1850229480761} a^{3} + \frac{456328819345}{1850229480761} a^{2} + \frac{4416015625}{1850229480761} a - \frac{640546220560}{1850229480761}$, $\frac{1}{1850229480761} a^{20} + \frac{142500}{1850229480761} a^{14} - \frac{641348734668}{1850229480761} a^{13} + \frac{6946875}{1850229480761} a^{12} - \frac{352925714101}{1850229480761} a^{11} + \frac{141075000}{1850229480761} a^{10} - \frac{728581855235}{1850229480761} a^{9} + \frac{1469531250}{1850229480761} a^{8} + \frac{857141306928}{1850229480761} a^{7} + \frac{8015625000}{1850229480761} a^{6} - \frac{682592524863}{1850229480761} a^{5} + \frac{21041015625}{1850229480761} a^{4} - \frac{187035559809}{1850229480761} a^{3} + \frac{20781250000}{1850229480761} a^{2} - \frac{60090969012}{1850229480761} a + \frac{3339843750}{1850229480761}$, $\frac{1}{1850229480761} a^{21} - \frac{788982181154}{1850229480761} a^{14} - \frac{3740625}{1850229480761} a^{13} + \frac{414109916445}{1850229480761} a^{12} - \frac{179550000}{1850229480761} a^{11} + \frac{12373696409}{1850229480761} a^{10} - \frac{3428906250}{1850229480761} a^{9} - \frac{140296237038}{1850229480761} a^{8} - \frac{32062500000}{1850229480761} a^{7} - \frac{263737409581}{1850229480761} a^{6} - \frac{147287109375}{1850229480761} a^{5} + \frac{592404997544}{1850229480761} a^{4} - \frac{290937500000}{1850229480761} a^{3} + \frac{451652357489}{1850229480761} a^{2} - \frac{163652343750}{1850229480761} a - \frac{915076790113}{1850229480761}$, $\frac{1}{1850229480761} a^{22} - \frac{4571875}{1850229480761} a^{14} + \frac{380430118643}{1850229480761} a^{13} - \frac{237737500}{1850229480761} a^{12} + \frac{852209243110}{1850229480761} a^{11} - \frac{5029062500}{1850229480761} a^{10} + \frac{458451936974}{1850229480761} a^{9} - \frac{53882812500}{1850229480761} a^{8} - \frac{747374475333}{1850229480761} a^{7} - \frac{300029296875}{1850229480761} a^{6} - \frac{698778886310}{1850229480761} a^{5} - \frac{800078125000}{1850229480761} a^{4} - \frac{637415570594}{1850229480761} a^{3} - \frac{800078125000}{1850229480761} a^{2} + \frac{153484570522}{1850229480761} a - \frac{129882812500}{1850229480761}$, $\frac{1}{1850229480761} a^{23} - \frac{125313668754}{1850229480761} a^{14} + \frac{105153125}{1850229480761} a^{13} + \frac{604504259779}{1850229480761} a^{12} + \frac{5257656250}{1850229480761} a^{11} + \frac{122368078035}{1850229480761} a^{10} + \frac{103275390625}{1850229480761} a^{9} + \frac{724960294353}{1850229480761} a^{8} - \frac{864418933886}{1850229480761} a^{7} + \frac{356417097687}{1850229480761} a^{6} + \frac{899990257228}{1850229480761} a^{5} + \frac{412931735048}{1850229480761} a^{4} - \frac{50248966305}{1850229480761} a^{3} + \frac{541303962194}{1850229480761} a^{2} - \frac{322905239158}{1850229480761} a + \frac{834342521060}{1850229480761}$, $\frac{1}{1850229480761} a^{24} + \frac{132825000}{1850229480761} a^{14} + \frac{751882012524}{1850229480761} a^{13} + \frac{7194687500}{1850229480761} a^{12} + \frac{843241698863}{1850229480761} a^{11} + \frac{156543750000}{1850229480761} a^{10} - \frac{802136979266}{1850229480761} a^{9} - \frac{138032215136}{1850229480761} a^{8} - \frac{195624856102}{1850229480761} a^{7} + \frac{434008846195}{1850229480761} a^{6} - \frac{425460886257}{1850229480761} a^{5} + \frac{246709144346}{1850229480761} a^{4} - \frac{257477770283}{1850229480761} a^{3} + \frac{510849769346}{1850229480761} a^{2} + \frac{76025614240}{1850229480761} a + \frac{623271507228}{1850229480761}$, $\frac{1}{1850229480761} a^{25} + \frac{58951101098}{1850229480761} a^{14} - \frac{2767187500}{1850229480761} a^{13} + \frac{444044398829}{1850229480761} a^{12} - \frac{142312500000}{1850229480761} a^{11} - \frac{678685845635}{1850229480761} a^{10} + \frac{846796852147}{1850229480761} a^{9} - \frac{26024427211}{1850229480761} a^{8} + \frac{81567211415}{1850229480761} a^{7} + \frac{761742115980}{1850229480761} a^{6} + \frac{616683759031}{1850229480761} a^{5} - \frac{593391015}{1850229480761} a^{4} + \frac{442190748823}{1850229480761} a^{3} + \frac{397131088325}{1850229480761} a^{2} + \frac{388251016152}{1850229480761} a - \frac{576384621063}{1850229480761}$, $\frac{1}{1850229480761} a^{26} - \frac{3597343750}{1850229480761} a^{14} - \frac{276829221999}{1850229480761} a^{13} - \frac{200423437500}{1850229480761} a^{12} - \frac{102140701343}{1850229480761} a^{11} - \frac{751253929103}{1850229480761} a^{10} - \frac{468843237666}{1850229480761} a^{9} + \frac{492719418047}{1850229480761} a^{8} + \frac{670935402801}{1850229480761} a^{7} - \frac{205709756067}{1850229480761} a^{6} - \frac{752027482519}{1850229480761} a^{5} + \frac{784594833098}{1850229480761} a^{4} + \frac{307599812742}{1850229480761} a^{3} - \frac{571415989075}{1850229480761} a^{2} - \frac{29631185595}{1850229480761} a - \frac{195850409230}{1850229480761}$, $\frac{1}{1850229480761} a^{27} - \frac{320616758615}{1850229480761} a^{14} + \frac{69377343750}{1850229480761} a^{13} + \frac{533190697059}{1850229480761} a^{12} - \frac{58148414647}{1850229480761} a^{11} + \frac{350704891540}{1850229480761} a^{10} + \frac{186035613310}{1850229480761} a^{9} + \frac{240606785382}{1850229480761} a^{8} - \frac{528306044834}{1850229480761} a^{7} - \frac{63868842930}{1850229480761} a^{6} + \frac{169782212581}{1850229480761} a^{5} - \frac{482253856563}{1850229480761} a^{4} - \frac{407944540608}{1850229480761} a^{3} - \frac{554390902036}{1850229480761} a^{2} + \frac{618599448462}{1850229480761} a + \frac{346023604178}{1850229480761}$

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:  $13$
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:  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

$C_{28}$ (as 28T1):

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

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.10755549.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 $28$ R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed