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Properties

Label 14.0.59945591630...8603.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{6}\cdot 13^{12}$
Root discriminant $35.94$
Ramified primes $3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (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![49, 49, -63, 154, 319, 22, 161, 96, -32, 14, 19, -18, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 7*x^12 - 18*x^11 + 19*x^10 + 14*x^9 - 32*x^8 + 96*x^7 + 161*x^6 + 22*x^5 + 319*x^4 + 154*x^3 - 63*x^2 + 49*x + 49)
 
gp: K = bnfinit(x^14 - 3*x^13 + 7*x^12 - 18*x^11 + 19*x^10 + 14*x^9 - 32*x^8 + 96*x^7 + 161*x^6 + 22*x^5 + 319*x^4 + 154*x^3 - 63*x^2 + 49*x + 49, 1)
 

Normalized defining polynomial

\( x^{14} - 3 x^{13} + 7 x^{12} - 18 x^{11} + 19 x^{10} + 14 x^{9} - 32 x^{8} + 96 x^{7} + 161 x^{6} + 22 x^{5} + 319 x^{4} + 154 x^{3} - 63 x^{2} + 49 x + 49 \)

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:  \(-5994559163049015798603=-\,3^{7}\cdot 7^{6}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.94$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} + \frac{2}{7} a^{5} + \frac{1}{7} a^{2}$, $\frac{1}{35} a^{9} + \frac{9}{35} a^{6} + \frac{1}{35} a^{3} + \frac{2}{5}$, $\frac{1}{105} a^{10} + \frac{1}{105} a^{9} + \frac{3}{35} a^{7} + \frac{44}{105} a^{6} + \frac{12}{35} a^{4} + \frac{1}{105} a^{3} - \frac{1}{3} a^{2} - \frac{1}{5} a + \frac{7}{15}$, $\frac{1}{105} a^{11} - \frac{1}{105} a^{9} - \frac{2}{35} a^{8} + \frac{1}{3} a^{7} - \frac{44}{105} a^{6} + \frac{2}{35} a^{5} - \frac{1}{3} a^{4} - \frac{12}{35} a^{3} - \frac{1}{105} a^{2} - \frac{1}{3} a - \frac{7}{15}$, $\frac{1}{5985} a^{12} + \frac{4}{1197} a^{11} - \frac{8}{1995} a^{10} - \frac{16}{5985} a^{9} + \frac{64}{1197} a^{8} + \frac{239}{5985} a^{7} + \frac{223}{855} a^{6} + \frac{431}{1197} a^{5} + \frac{221}{5985} a^{4} + \frac{306}{665} a^{3} - \frac{1}{57} a^{2} - \frac{413}{855} a - \frac{41}{171}$, $\frac{1}{784035} a^{13} + \frac{58}{784035} a^{12} + \frac{2104}{784035} a^{11} + \frac{487}{156807} a^{10} + \frac{2}{399} a^{9} + \frac{1136}{37335} a^{8} + \frac{46486}{156807} a^{7} + \frac{1897}{7467} a^{6} + \frac{171544}{784035} a^{5} + \frac{33626}{156807} a^{4} - \frac{3834}{17423} a^{3} - \frac{357659}{784035} a^{2} - \frac{25133}{112005} a - \frac{1421}{5895}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{167}{13755} a^{13} + \frac{221}{4585} a^{12} - \frac{247}{1965} a^{11} + \frac{4433}{13755} a^{10} - \frac{52}{105} a^{9} + \frac{2509}{13755} a^{8} + \frac{1664}{4585} a^{7} - \frac{21073}{13755} a^{6} - \frac{2483}{4585} a^{5} + \frac{1859}{1965} a^{4} - \frac{18304}{4585} a^{3} + \frac{31174}{13755} a^{2} + \frac{689}{655} a - \frac{973}{1965} \) (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:  \( 1933477.60794 \) (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

\(\Q(\sqrt{-3}) \), 7.1.1655595487.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$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}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$