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Properties

Label 14.0.44931703384...1103.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,463^{13}$
Root discriminant $298.66$
Ramified prime $463$
Class number $11368$ (GRH)
Class group $[2, 2, 2842]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2026251923, -1637661044, 591306086, -321562607, 168859238, -48211158, 7843772, -900107, 123934, -9712, -540, -71, 17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 17*x^12 - 71*x^11 - 540*x^10 - 9712*x^9 + 123934*x^8 - 900107*x^7 + 7843772*x^6 - 48211158*x^5 + 168859238*x^4 - 321562607*x^3 + 591306086*x^2 - 1637661044*x + 2026251923)
 
gp: K = bnfinit(x^14 - x^13 + 17*x^12 - 71*x^11 - 540*x^10 - 9712*x^9 + 123934*x^8 - 900107*x^7 + 7843772*x^6 - 48211158*x^5 + 168859238*x^4 - 321562607*x^3 + 591306086*x^2 - 1637661044*x + 2026251923, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 17 x^{12} - 71 x^{11} - 540 x^{10} - 9712 x^{9} + 123934 x^{8} - 900107 x^{7} + 7843772 x^{6} - 48211158 x^{5} + 168859238 x^{4} - 321562607 x^{3} + 591306086 x^{2} - 1637661044 x + 2026251923 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-44931703384098017538267680344521103=-\,463^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $298.66$
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:  $463$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(463\)
Dirichlet character group:    $\lbrace$$\chi_{463}(1,·)$, $\chi_{463}(34,·)$, $\chi_{463}(230,·)$, $\chi_{463}(233,·)$, $\chi_{463}(429,·)$, $\chi_{463}(462,·)$, $\chi_{463}(177,·)$, $\chi_{463}(51,·)$, $\chi_{463}(308,·)$, $\chi_{463}(118,·)$, $\chi_{463}(345,·)$, $\chi_{463}(155,·)$, $\chi_{463}(412,·)$, $\chi_{463}(286,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17956525113107271417578514459637783728662633780145668053} a^{13} - \frac{5173915121217981566351072345321551962694373611525110285}{17956525113107271417578514459637783728662633780145668053} a^{12} - \frac{7253454099609977956763895768194666838547726049141261232}{17956525113107271417578514459637783728662633780145668053} a^{11} + \frac{6530791167880084687031818527547461775464011372558956320}{17956525113107271417578514459637783728662633780145668053} a^{10} - \frac{5807546098025729252611581919979626002278017859811205210}{17956525113107271417578514459637783728662633780145668053} a^{9} + \frac{8846337914653136036211436332296002762872900529537384252}{17956525113107271417578514459637783728662633780145668053} a^{8} + \frac{3080116651017228066680786338845887534078815681994764515}{17956525113107271417578514459637783728662633780145668053} a^{7} + \frac{4139231924879246327638871002928697109726689093167245132}{17956525113107271417578514459637783728662633780145668053} a^{6} + \frac{4250567385558844058292896881994585032600708801353548094}{17956525113107271417578514459637783728662633780145668053} a^{5} + \frac{775809420569211250058200889507840796649165959505447919}{17956525113107271417578514459637783728662633780145668053} a^{4} + \frac{1322056732729035736404053911143816321231881675282902255}{17956525113107271417578514459637783728662633780145668053} a^{3} - \frac{242222169967161211764879138161867630381526966570133361}{17956525113107271417578514459637783728662633780145668053} a^{2} + \frac{720247804531336282662672859019338135155353098904190216}{17956525113107271417578514459637783728662633780145668053} a + \frac{3297312018739794847373893437764309795110352651245964472}{17956525113107271417578514459637783728662633780145668053}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2842}$, which has order $11368$ (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:  $6$
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:  \( 12278961.491848389 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-463}) \), 7.7.9851127637605409.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
463Data not computed