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Properties

Label 14.0.48052913294...9056.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 3^{7}\cdot 7^{25}$
Root discriminant $111.87$
Ramified primes $2, 3, 7$
Class number $14284$ (GRH)
Class group $[2, 7142]$ (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![15309, 0, 142884, 0, 357210, 0, 305613, 0, 64827, 0, 4851, 0, 126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 126*x^12 + 4851*x^10 + 64827*x^8 + 305613*x^6 + 357210*x^4 + 142884*x^2 + 15309)
 
gp: K = bnfinit(x^14 + 126*x^12 + 4851*x^10 + 64827*x^8 + 305613*x^6 + 357210*x^4 + 142884*x^2 + 15309, 1)
 

Normalized defining polynomial

\( x^{14} + 126 x^{12} + 4851 x^{10} + 64827 x^{8} + 305613 x^{6} + 357210 x^{4} + 142884 x^{2} + 15309 \)

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:  \(-48052913294624214844455469056=-\,2^{14}\cdot 3^{7}\cdot 7^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.87$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(588=2^{2}\cdot 3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{588}(1,·)$, $\chi_{588}(419,·)$, $\chi_{588}(421,·)$, $\chi_{588}(167,·)$, $\chi_{588}(169,·)$, $\chi_{588}(587,·)$, $\chi_{588}(335,·)$, $\chi_{588}(337,·)$, $\chi_{588}(83,·)$, $\chi_{588}(85,·)$, $\chi_{588}(503,·)$, $\chi_{588}(505,·)$, $\chi_{588}(251,·)$, $\chi_{588}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{364743} a^{10} - \frac{302}{121581} a^{8} - \frac{236}{40527} a^{6} - \frac{101}{4503} a^{4} - \frac{55}{4503} a^{2} + \frac{638}{1501}$, $\frac{1}{364743} a^{11} - \frac{302}{121581} a^{9} - \frac{236}{40527} a^{7} - \frac{101}{4503} a^{5} - \frac{55}{4503} a^{3} + \frac{638}{1501} a$, $\frac{1}{33921099} a^{12} + \frac{1}{1256337} a^{10} - \frac{1366}{1256337} a^{8} + \frac{5854}{1256337} a^{6} - \frac{303}{46531} a^{4} + \frac{3517}{46531} a^{2} - \frac{16225}{46531}$, $\frac{1}{33921099} a^{13} + \frac{1}{1256337} a^{11} - \frac{1366}{1256337} a^{9} + \frac{5854}{1256337} a^{7} - \frac{303}{46531} a^{5} + \frac{3517}{46531} a^{3} - \frac{16225}{46531} a$

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

Class group and class number

$C_{2}\times C_{7142}$, which has order $14284$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-21}) \), 7.7.13841287201.1

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 R R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.25.75$x^{14} + 112 x^{13} + 28 x^{12} - 98 x^{11} + 147 x^{10} + 98 x^{9} + 49 x^{8} - 161 x^{7} + 49 x^{6} + 147 x^{5} + 147 x^{2} - 147 x - 126$$14$$1$$25$$C_{14}$$[2]_{2}$