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Properties

Label 14.0.64646372906...9287.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,967^{13}$
Root discriminant $591.81$
Ramified prime $967$
Class number $181192$ (GRH)
Class group $[2, 2, 45298]$ (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![7873416461, -3237370650, -65709808, 683421421, 54692142, -206239922, 106373950, -26254395, 3261324, -96498, -16886, 957, 35, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 35*x^12 + 957*x^11 - 16886*x^10 - 96498*x^9 + 3261324*x^8 - 26254395*x^7 + 106373950*x^6 - 206239922*x^5 + 54692142*x^4 + 683421421*x^3 - 65709808*x^2 - 3237370650*x + 7873416461)
 
gp: K = bnfinit(x^14 - x^13 + 35*x^12 + 957*x^11 - 16886*x^10 - 96498*x^9 + 3261324*x^8 - 26254395*x^7 + 106373950*x^6 - 206239922*x^5 + 54692142*x^4 + 683421421*x^3 - 65709808*x^2 - 3237370650*x + 7873416461, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 35 x^{12} + 957 x^{11} - 16886 x^{10} - 96498 x^{9} + 3261324 x^{8} - 26254395 x^{7} + 106373950 x^{6} - 206239922 x^{5} + 54692142 x^{4} + 683421421 x^{3} - 65709808 x^{2} - 3237370650 x + 7873416461 \)

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:  \(-646463729067293446729771507117629989287=-\,967^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $591.81$
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:  $967$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(967\)
Dirichlet character group:    $\lbrace$$\chi_{967}(1,·)$, $\chi_{967}(226,·)$, $\chi_{967}(261,·)$, $\chi_{967}(870,·)$, $\chi_{967}(97,·)$, $\chi_{967}(648,·)$, $\chi_{967}(706,·)$, $\chi_{967}(175,·)$, $\chi_{967}(536,·)$, $\chi_{967}(319,·)$, $\chi_{967}(792,·)$, $\chi_{967}(966,·)$, $\chi_{967}(431,·)$, $\chi_{967}(741,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6893} a^{12} - \frac{2813}{6893} a^{11} - \frac{838}{6893} a^{10} - \frac{1945}{6893} a^{9} + \frac{3141}{6893} a^{8} + \frac{1349}{6893} a^{7} - \frac{2157}{6893} a^{6} - \frac{473}{6893} a^{5} + \frac{230}{6893} a^{4} + \frac{3236}{6893} a^{3} - \frac{3038}{6893} a^{2} - \frac{1240}{6893} a + \frac{6}{113}$, $\frac{1}{644584180745870782164520926853011313969689170291801728065797} a^{13} + \frac{4455192274822787934826618655167460888737121442450120087}{644584180745870782164520926853011313969689170291801728065797} a^{12} + \frac{69363662467496189239035212227277608967005242545426160619628}{644584180745870782164520926853011313969689170291801728065797} a^{11} - \frac{193744589433958309594390076281516487252999093780814141894592}{644584180745870782164520926853011313969689170291801728065797} a^{10} + \frac{220656490716998570399208222896879638857814352274498685596038}{644584180745870782164520926853011313969689170291801728065797} a^{9} - \frac{200805636145032112588561062437264360955281605102798860161593}{644584180745870782164520926853011313969689170291801728065797} a^{8} - \frac{3063588612522355877012117002348167073664121746502075901457}{8829920284190010714582478450041250876297111921805503124189} a^{7} + \frac{164232645950861181891765921660377788005343232736505766459193}{644584180745870782164520926853011313969689170291801728065797} a^{6} + \frac{207775005853585706836174306938882849354592095314363824511018}{644584180745870782164520926853011313969689170291801728065797} a^{5} - \frac{258826876791330249955232022737869045305996244130904469103674}{644584180745870782164520926853011313969689170291801728065797} a^{4} - \frac{177432544993138358903826620895751606794954643544892453638293}{644584180745870782164520926853011313969689170291801728065797} a^{3} - \frac{205587999489995968447784863162776174329308728905406085880464}{644584180745870782164520926853011313969689170291801728065797} a^{2} - \frac{37918301019800597487078066278192143195609321918663081866161}{644584180745870782164520926853011313969689170291801728065797} a - \frac{1356589821242649105543153823474185399112128610426150614527}{10566953782719193150238047981196906786388347053963962755177}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{45298}$, which has order $181192$ (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:  \( 193748644.5997488 \) (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

\(\Q(\sqrt{-967}) \), 7.7.817633815294109969.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
967Data not computed