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Properties

Label 14.10.1273417700...5088.2
Degree $14$
Signature $[10, 2]$
Discriminant $2^{16}\cdot 13^{7}\cdot 17597207^{2}$
Root discriminant $86.31$
Ramified primes $2, 13, 17597207$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T52

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 448, 800, -512, -2296, -700, 1799, 968, -618, -384, 123, 64, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 16*x^12 + 64*x^11 + 123*x^10 - 384*x^9 - 618*x^8 + 968*x^7 + 1799*x^6 - 700*x^5 - 2296*x^4 - 512*x^3 + 800*x^2 + 448*x + 64)
 
gp: K = bnfinit(x^14 - 4*x^13 - 16*x^12 + 64*x^11 + 123*x^10 - 384*x^9 - 618*x^8 + 968*x^7 + 1799*x^6 - 700*x^5 - 2296*x^4 - 512*x^3 + 800*x^2 + 448*x + 64, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 16 x^{12} + 64 x^{11} + 123 x^{10} - 384 x^{9} - 618 x^{8} + 968 x^{7} + 1799 x^{6} - 700 x^{5} - 2296 x^{4} - 512 x^{3} + 800 x^{2} + 448 x + 64 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1273417700659086943252185088=2^{16}\cdot 13^{7}\cdot 17597207^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.31$
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:  $2, 13, 17597207$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a$, $\frac{1}{48} a^{11} + \frac{1}{16} a^{7} - \frac{1}{12} a^{6} - \frac{1}{24} a^{5} - \frac{1}{2} a^{4} - \frac{17}{48} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{96} a^{12} - \frac{1}{12} a^{9} - \frac{5}{96} a^{8} - \frac{1}{24} a^{7} - \frac{5}{48} a^{6} - \frac{1}{4} a^{5} + \frac{13}{32} a^{4} + \frac{1}{6} a^{3} - \frac{5}{12} a^{2} + \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{192} a^{13} - \frac{1}{24} a^{10} - \frac{5}{192} a^{9} + \frac{1}{16} a^{8} - \frac{5}{96} a^{7} + \frac{5}{24} a^{6} + \frac{7}{192} a^{5} + \frac{5}{12} a^{4} - \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{3}$

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:  $11$
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:  \( 915328456.218 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T52:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 56448
The 27 conjugacy class representatives for [L(7)^2]2=L(7)wr2
Character table for [L(7)^2]2=L(7)wr2 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \)

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

Sibling fields

Degree 16 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 sibling: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17597207Data not computed