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Properties

Label 14.0.12382566156...1111.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,1031^{7}$
Root discriminant $32.11$
Ramified prime $1031$
Class number $5$
Class group $[5]$
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![50519, 0, 29709, 0, 3950, 0, -281, 0, -28, 0, 31, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 10*x^12 + 31*x^10 - 28*x^8 - 281*x^6 + 3950*x^4 + 29709*x^2 + 50519)
 
gp: K = bnfinit(x^14 + 10*x^12 + 31*x^10 - 28*x^8 - 281*x^6 + 3950*x^4 + 29709*x^2 + 50519, 1)
 

Normalized defining polynomial

\( x^{14} + 10 x^{12} + 31 x^{10} - 28 x^{8} - 281 x^{6} + 3950 x^{4} + 29709 x^{2} + 50519 \)

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:  \(-1238256615687209381111=-\,1031^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.11$
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:  $1031$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{8} - \frac{1}{2} a^{6} + \frac{1}{26} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{6}{13}$, $\frac{1}{26} a^{9} - \frac{6}{13} a^{5} - \frac{1}{2} a^{2} + \frac{1}{26} a - \frac{1}{2}$, $\frac{1}{26} a^{10} - \frac{6}{13} a^{6} - \frac{1}{2} a^{3} + \frac{1}{26} a^{2} - \frac{1}{2} a$, $\frac{1}{26} a^{11} + \frac{1}{26} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{6}{13} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1274891722} a^{12} + \frac{169373}{98068594} a^{10} - \frac{1006321}{637445861} a^{8} + \frac{15471207}{49034297} a^{6} - \frac{1}{2} a^{5} + \frac{297789159}{1274891722} a^{4} - \frac{21971785}{98068594} a^{2} - \frac{1}{2} a + \frac{37021519}{637445861}$, $\frac{1}{8924242054} a^{13} + \frac{1970621}{343240079} a^{11} + \frac{145090249}{8924242054} a^{9} - \frac{23060415}{98068594} a^{7} - \frac{1}{2} a^{6} + \frac{1497337747}{4462121027} a^{5} - \frac{1}{2} a^{4} + \frac{128902975}{686480158} a^{3} - \frac{1}{2} a^{2} - \frac{208149966}{4462121027} a$

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

Class group and class number

$C_{5}$, which has order $5$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18383.6173323 \)
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

\(\Q(\sqrt{-1031}) \), 7.1.1095912791.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.1095912791.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
1031Data not computed