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Properties

Label 14.0.18849407496...2759.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7879^{7}$
Root discriminant $88.76$
Ramified prime $7879$
Class number $1183$ (GRH)
Class group $[13, 91]$ (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![6626239, 0, 6367700, 0, 110594, 0, 24852, 0, 1227, 0, 236, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 32*x^12 + 236*x^10 + 1227*x^8 + 24852*x^6 + 110594*x^4 + 6367700*x^2 + 6626239)
 
gp: K = bnfinit(x^14 + 32*x^12 + 236*x^10 + 1227*x^8 + 24852*x^6 + 110594*x^4 + 6367700*x^2 + 6626239, 1)
 

Normalized defining polynomial

\( x^{14} + 32 x^{12} + 236 x^{10} + 1227 x^{8} + 24852 x^{6} + 110594 x^{4} + 6367700 x^{2} + 6626239 \)

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:  \(-1884940749689661095518792759=-\,7879^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.76$
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:  $7879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{18} a^{8} + \frac{1}{9} a^{6} - \frac{1}{2} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{2} - \frac{1}{2} a - \frac{1}{9}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{7} - \frac{1}{6} a^{6} + \frac{4}{9} a^{5} - \frac{1}{6} a^{4} - \frac{2}{9} a^{3} - \frac{1}{6} a^{2} + \frac{2}{9} a - \frac{1}{6}$, $\frac{1}{54} a^{10} - \frac{1}{27} a^{6} - \frac{5}{54} a^{4} - \frac{1}{2} a^{3} - \frac{23}{54}$, $\frac{1}{54} a^{11} - \frac{1}{27} a^{7} - \frac{5}{54} a^{5} - \frac{1}{2} a^{4} - \frac{23}{54} a$, $\frac{1}{363037840780642146} a^{12} + \frac{1605289400999749}{363037840780642146} a^{10} + \frac{4583539222435483}{363037840780642146} a^{8} - \frac{4897293197643379}{51862548682948878} a^{6} - \frac{51002431685403791}{363037840780642146} a^{4} - \frac{1}{2} a^{3} + \frac{99766686874485091}{363037840780642146} a^{2} - \frac{1}{2} a - \frac{43014485971483925}{363037840780642146}$, $\frac{1}{10528097382638622234} a^{13} + \frac{24332875121467421}{5264048691319311117} a^{11} + \frac{72882460874120381}{5264048691319311117} a^{9} - \frac{47155666198564687}{1504013911805517462} a^{7} - \frac{1}{6} a^{6} - \frac{2422223257292681989}{5264048691319311117} a^{5} + \frac{1}{3} a^{4} - \frac{2522173274319041519}{10528097382638622234} a^{3} + \frac{1}{3} a^{2} + \frac{1232317892302957301}{5264048691319311117} a + \frac{1}{3}$

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

Class group and class number

$C_{13}\times C_{91}$, which has order $1183$ (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:  \( 207826.01611 \) (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

\(\Q(\sqrt{-7879}) \), 7.1.489117612439.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.489117612439.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
7879Data not computed