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Properties

Label 14.0.74811057150...3936.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 7^{25}\cdot 23^{7}$
Root discriminant $309.74$
Ramified primes $2, 7, 23$
Class number $14396048$ (GRH)
Class group $[2, 7198024]$ (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![23833778129, 0, 29015034244, 0, 9461424210, 0, 1055840093, 0, 29212967, 0, 285131, 0, 966, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 966*x^12 + 285131*x^10 + 29212967*x^8 + 1055840093*x^6 + 9461424210*x^4 + 29015034244*x^2 + 23833778129)
 
gp: K = bnfinit(x^14 + 966*x^12 + 285131*x^10 + 29212967*x^8 + 1055840093*x^6 + 9461424210*x^4 + 29015034244*x^2 + 23833778129, 1)
 

Normalized defining polynomial

\( x^{14} + 966 x^{12} + 285131 x^{10} + 29212967 x^{8} + 1055840093 x^{6} + 9461424210 x^{4} + 29015034244 x^{2} + 23833778129 \)

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:  \(-74811057150444048927662152674983936=-\,2^{14}\cdot 7^{25}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $309.74$
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:  $2, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4508=2^{2}\cdot 7^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{4508}(1,·)$, $\chi_{4508}(643,·)$, $\chi_{4508}(645,·)$, $\chi_{4508}(1287,·)$, $\chi_{4508}(1289,·)$, $\chi_{4508}(1931,·)$, $\chi_{4508}(1933,·)$, $\chi_{4508}(2575,·)$, $\chi_{4508}(2577,·)$, $\chi_{4508}(3219,·)$, $\chi_{4508}(3221,·)$, $\chi_{4508}(3863,·)$, $\chi_{4508}(3865,·)$, $\chi_{4508}(4507,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{23} a^{2}$, $\frac{1}{23} a^{3}$, $\frac{1}{529} a^{4}$, $\frac{1}{529} a^{5}$, $\frac{1}{12167} a^{6}$, $\frac{1}{12167} a^{7}$, $\frac{1}{279841} a^{8}$, $\frac{1}{279841} a^{9}$, $\frac{1}{9660950843} a^{10} - \frac{302}{420041341} a^{8} - \frac{236}{18262667} a^{6} - \frac{303}{794029} a^{4} - \frac{55}{34523} a^{2} + \frac{638}{1501}$, $\frac{1}{9660950843} a^{11} - \frac{302}{420041341} a^{9} - \frac{236}{18262667} a^{7} - \frac{303}{794029} a^{5} - \frac{55}{34523} a^{3} + \frac{638}{1501} a$, $\frac{1}{6888257951059} a^{12} + \frac{9}{299489476133} a^{10} - \frac{4098}{13021281571} a^{8} + \frac{5854}{566142677} a^{6} - \frac{2727}{24614899} a^{4} + \frac{10551}{1070213} a^{2} - \frac{16225}{46531}$, $\frac{1}{6888257951059} a^{13} + \frac{9}{299489476133} a^{11} - \frac{4098}{13021281571} a^{9} + \frac{5854}{566142677} a^{7} - \frac{2727}{24614899} a^{5} + \frac{10551}{1070213} a^{3} - \frac{16225}{46531} a$

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

Class group and class number

$C_{2}\times C_{7198024}$, which has order $14396048$ (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

\(\Q(\sqrt{-161}) \), 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$23$23.14.7.1$x^{14} - 24334 x^{8} + 148035889 x^{2} - 217908828608$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$