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Properties

Label 14.0.65111179469...1936.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 11^{7}\cdot 13^{8}$
Root discriminant $26.02$
Ramified primes $2, 11, 13$
Class number $2$
Class group $[2]$
Galois group 14T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44, 0, -107, 0, 106, 0, -69, 0, 32, 0, 1, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^12 + x^10 + 32*x^8 - 69*x^6 + 106*x^4 - 107*x^2 + 44)
 
gp: K = bnfinit(x^14 - 5*x^12 + x^10 + 32*x^8 - 69*x^6 + 106*x^4 - 107*x^2 + 44, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{12} + x^{10} + 32 x^{8} - 69 x^{6} + 106 x^{4} - 107 x^{2} + 44 \)

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:  \(-65111179469128871936=-\,2^{12}\cdot 11^{7}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.02$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{10539} a^{12} + \frac{463}{3513} a^{10} - \frac{2909}{10539} a^{8} + \frac{2401}{10539} a^{6} - \frac{4477}{10539} a^{4} - \frac{1744}{10539} a^{2} + \frac{3266}{10539}$, $\frac{1}{42156} a^{13} - \frac{1}{21078} a^{12} - \frac{1879}{14052} a^{11} - \frac{463}{7026} a^{10} + \frac{4117}{42156} a^{9} + \frac{2909}{21078} a^{8} - \frac{278}{10539} a^{7} + \frac{4069}{10539} a^{6} - \frac{18529}{42156} a^{5} + \frac{4477}{21078} a^{4} + \frac{9667}{21078} a^{3} + \frac{872}{10539} a^{2} - \frac{247}{42156} a + \frac{7273}{21078}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 59972.4555344 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2688
The 20 conjugacy class representatives for 1/2[2^7]F_42(7)
Character table for 1/2[2^7]F_42(7)

Intermediate fields

7.1.38014691.1

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

Sibling fields

Degree 14 sibling: data not computed
Degree 16 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 siblings: 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.12.12.13$x^{12} - 18 x^{10} - 13 x^{8} - 44 x^{6} + 55 x^{4} + 62 x^{2} + 21$$2$$6$$12$12T105$[2, 2, 2, 2, 2]^{6}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$