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Properties

Label 14.0.23931305644896387.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 149^{6}$
Root discriminant $14.79$
Ramified primes $3, 149$
Class number $1$
Class group Trivial
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![3, -12, 58, -113, 135, -164, 174, -118, 60, -34, 9, 7, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 3*x^12 + 7*x^11 + 9*x^10 - 34*x^9 + 60*x^8 - 118*x^7 + 174*x^6 - 164*x^5 + 135*x^4 - 113*x^3 + 58*x^2 - 12*x + 3)
 
gp: K = bnfinit(x^14 - 2*x^13 - 3*x^12 + 7*x^11 + 9*x^10 - 34*x^9 + 60*x^8 - 118*x^7 + 174*x^6 - 164*x^5 + 135*x^4 - 113*x^3 + 58*x^2 - 12*x + 3, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 3 x^{12} + 7 x^{11} + 9 x^{10} - 34 x^{9} + 60 x^{8} - 118 x^{7} + 174 x^{6} - 164 x^{5} + 135 x^{4} - 113 x^{3} + 58 x^{2} - 12 x + 3 \)

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:  \(-23931305644896387=-\,3^{7}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.79$
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:  $3, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{15} a^{10} - \frac{2}{15} a^{9} - \frac{2}{15} a^{8} + \frac{2}{5} a^{6} - \frac{2}{15} a^{5} - \frac{2}{15} a^{4} - \frac{4}{15} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{11} - \frac{2}{5} a^{9} - \frac{4}{15} a^{8} + \frac{2}{5} a^{7} - \frac{1}{3} a^{6} - \frac{2}{5} a^{5} - \frac{4}{15} a^{4} - \frac{4}{15} a^{3} - \frac{1}{3} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{9} - \frac{2}{5} a^{8} - \frac{1}{3} a^{7} - \frac{1}{15} a^{5} - \frac{1}{15} a^{4} - \frac{1}{3} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{8205} a^{13} + \frac{227}{8205} a^{12} + \frac{1}{547} a^{11} + \frac{32}{1641} a^{10} - \frac{923}{2735} a^{8} + \frac{424}{1641} a^{7} + \frac{691}{1641} a^{6} - \frac{575}{1641} a^{5} + \frac{3331}{8205} a^{4} - \frac{245}{1641} a^{3} + \frac{104}{1641} a^{2} + \frac{1058}{2735} a - \frac{1137}{2735}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1063}{8205} a^{13} - \frac{207}{2735} a^{12} + \frac{4294}{8205} a^{11} + \frac{3319}{8205} a^{10} - \frac{7}{5} a^{9} + \frac{4963}{8205} a^{8} - \frac{6902}{2735} a^{7} + \frac{39824}{8205} a^{6} - \frac{7622}{8205} a^{5} - \frac{8327}{8205} a^{4} - \frac{6796}{8205} a^{3} - \frac{18341}{8205} a^{2} + \frac{9277}{2735} a + \frac{308}{2735} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1043.56930973 \)
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

\(\Q(\sqrt{-3}) \), 7.1.89314623.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$