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Properties

Label 14.0.89469990354...3683.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 587^{6}$
Root discriminant $26.61$
Ramified primes $3, 587$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $D_{14}$ (as 14T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![171, 405, 528, 423, 361, 162, 118, -17, 82, -83, 67, -44, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 19*x^12 - 44*x^11 + 67*x^10 - 83*x^9 + 82*x^8 - 17*x^7 + 118*x^6 + 162*x^5 + 361*x^4 + 423*x^3 + 528*x^2 + 405*x + 171)
 
gp: K = bnfinit(x^14 - 6*x^13 + 19*x^12 - 44*x^11 + 67*x^10 - 83*x^9 + 82*x^8 - 17*x^7 + 118*x^6 + 162*x^5 + 361*x^4 + 423*x^3 + 528*x^2 + 405*x + 171, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 19 x^{12} - 44 x^{11} + 67 x^{10} - 83 x^{9} + 82 x^{8} - 17 x^{7} + 118 x^{6} + 162 x^{5} + 361 x^{4} + 423 x^{3} + 528 x^{2} + 405 x + 171 \)

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:  \(-89469990354509983683=-\,3^{7}\cdot 587^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.61$
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:  $3, 587$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{1}{27} a^{8} - \frac{2}{27} a^{7} - \frac{1}{27} a^{6} - \frac{5}{27} a^{5} + \frac{1}{27} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{81} a^{11} + \frac{1}{81} a^{9} + \frac{2}{81} a^{8} + \frac{7}{81} a^{7} - \frac{10}{81} a^{6} + \frac{1}{27} a^{5} - \frac{25}{81} a^{4} + \frac{1}{27} a^{2} + \frac{2}{9}$, $\frac{1}{729} a^{12} - \frac{1}{729} a^{11} + \frac{1}{729} a^{10} - \frac{26}{729} a^{9} + \frac{5}{729} a^{8} + \frac{37}{729} a^{7} + \frac{40}{729} a^{6} + \frac{350}{729} a^{5} - \frac{272}{729} a^{4} + \frac{1}{243} a^{3} - \frac{73}{243} a^{2} - \frac{34}{81} a - \frac{11}{81}$, $\frac{1}{308367} a^{13} - \frac{134}{308367} a^{12} - \frac{595}{308367} a^{11} + \frac{838}{102789} a^{10} + \frac{3139}{308367} a^{9} + \frac{2936}{308367} a^{8} + \frac{12656}{102789} a^{7} + \frac{44764}{308367} a^{6} - \frac{113080}{308367} a^{5} - \frac{2210}{6561} a^{4} - \frac{2825}{102789} a^{3} - \frac{32918}{102789} a^{2} - \frac{9529}{34263} a + \frac{7862}{34263}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( \frac{29}{34263} a^{13} - \frac{173}{34263} a^{12} + \frac{605}{34263} a^{11} - \frac{2059}{34263} a^{10} + \frac{5914}{34263} a^{9} - \frac{14308}{34263} a^{8} + \frac{27827}{34263} a^{7} - \frac{38054}{34263} a^{6} + \frac{43016}{34263} a^{5} - \frac{185}{243} a^{4} + \frac{5542}{11421} a^{3} - \frac{785}{3807} a^{2} - \frac{310}{3807} a - \frac{73}{423} \) (order $6$)
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:  \( 65193.8249252 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{14}$ (as 14T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 7.7.5461074081.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 14 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
587Data not computed