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Properties

Label 14.0.10257069636...1047.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,31^{7}\cdot 233^{7}$
Root discriminant $84.99$
Ramified primes $31, 233$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5265567, 0, 1347921, 0, -65016, 0, -36368, 0, -1426, 0, 312, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 32*x^12 + 312*x^10 - 1426*x^8 - 36368*x^6 - 65016*x^4 + 1347921*x^2 + 5265567)
 
gp: K = bnfinit(x^14 + 32*x^12 + 312*x^10 - 1426*x^8 - 36368*x^6 - 65016*x^4 + 1347921*x^2 + 5265567, 1)
 

Normalized defining polynomial

\( x^{14} + 32 x^{12} + 312 x^{10} - 1426 x^{8} - 36368 x^{6} - 65016 x^{4} + 1347921 x^{2} + 5265567 \)

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:  \(-1025706963691171311737151047=-\,31^{7}\cdot 233^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.99$
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:  $31, 233$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{27} a^{6} - \frac{2}{27} a^{4} + \frac{1}{27} a^{2} - \frac{1}{3}$, $\frac{1}{54} a^{7} - \frac{1}{27} a^{5} - \frac{4}{27} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{54} a^{8} + \frac{1}{9} a^{4} + \frac{11}{54} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{162} a^{9} + \frac{4}{27} a^{5} - \frac{7}{162} a^{3} - \frac{1}{2} a^{2} + \frac{2}{9} a$, $\frac{1}{162} a^{10} - \frac{13}{162} a^{4} - \frac{1}{6} a^{3} + \frac{11}{27} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{162} a^{11} - \frac{13}{162} a^{5} - \frac{1}{6} a^{4} + \frac{2}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{388246671990} a^{12} + \frac{39624247}{77649334398} a^{10} + \frac{292022399}{129415557330} a^{8} + \frac{89508889}{77649334398} a^{6} - \frac{1}{6} a^{5} + \frac{24438646477}{388246671990} a^{4} - \frac{1}{6} a^{3} - \frac{3346761817}{8627703822} a^{2} - \frac{1}{6} a - \frac{86770447}{2396584395}$, $\frac{1}{1164740015970} a^{13} - \frac{219846316}{116474001597} a^{11} + \frac{292022399}{388246671990} a^{9} + \frac{763729763}{116474001597} a^{7} - \frac{1}{54} a^{6} - \frac{44100410044}{582370007985} a^{5} + \frac{1}{27} a^{4} + \frac{487773215}{25883111466} a^{3} - \frac{1}{54} a^{2} + \frac{2223043501}{14379506370} a + \frac{1}{6}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 807224869.359 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-7223}) \), 7.1.376836398567.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.376836398567.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
233Data not computed