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Properties

Label 14.6.13290133788...0000.1
Degree $14$
Signature $[6, 4]$
Discriminant $2^{14}\cdot 3^{7}\cdot 5^{7}\cdot 7^{15}$
Root discriminant $62.31$
Ramified primes $2, 3, 5, 7$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 14T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -98, -742, -3178, -7210, -9254, -6594, -1806, 896, 896, 196, -56, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 28*x^12 - 56*x^11 + 196*x^10 + 896*x^9 + 896*x^8 - 1806*x^7 - 6594*x^6 - 9254*x^5 - 7210*x^4 - 3178*x^3 - 742*x^2 - 98*x - 14)
 
gp: K = bnfinit(x^14 - 28*x^12 - 56*x^11 + 196*x^10 + 896*x^9 + 896*x^8 - 1806*x^7 - 6594*x^6 - 9254*x^5 - 7210*x^4 - 3178*x^3 - 742*x^2 - 98*x - 14, 1)
 

Normalized defining polynomial

\( x^{14} - 28 x^{12} - 56 x^{11} + 196 x^{10} + 896 x^{9} + 896 x^{8} - 1806 x^{7} - 6594 x^{6} - 9254 x^{5} - 7210 x^{4} - 3178 x^{3} - 742 x^{2} - 98 x - 14 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13290133788474036480000000=2^{14}\cdot 3^{7}\cdot 5^{7}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.31$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{33} a^{8} - \frac{2}{33} a^{7} + \frac{4}{33} a^{6} + \frac{1}{11} a^{5} + \frac{1}{3} a^{4} + \frac{2}{33} a^{3} + \frac{7}{33} a^{2} - \frac{5}{11} a + \frac{10}{33}$, $\frac{1}{33} a^{9} + \frac{1}{3} a^{6} - \frac{16}{33} a^{5} - \frac{3}{11} a^{4} + \frac{1}{3} a^{3} - \frac{1}{33} a^{2} + \frac{13}{33} a - \frac{13}{33}$, $\frac{1}{33} a^{10} - \frac{16}{33} a^{6} + \frac{13}{33} a^{5} - \frac{1}{3} a^{4} - \frac{1}{33} a^{3} - \frac{3}{11} a^{2} - \frac{2}{33} a - \frac{1}{3}$, $\frac{1}{33} a^{11} - \frac{5}{33} a^{7} + \frac{13}{33} a^{6} - \frac{4}{11} a^{4} - \frac{3}{11} a^{3} - \frac{13}{33} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{33} a^{12} + \frac{1}{11} a^{7} - \frac{13}{33} a^{6} + \frac{1}{11} a^{5} + \frac{13}{33} a^{4} - \frac{1}{11} a^{3} + \frac{13}{33} a^{2} + \frac{2}{33} a - \frac{16}{33}$, $\frac{1}{33} a^{13} + \frac{4}{33} a^{7} - \frac{3}{11} a^{6} + \frac{5}{11} a^{5} - \frac{14}{33} a^{4} + \frac{7}{33} a^{3} + \frac{1}{11} a^{2} - \frac{5}{11} a + \frac{14}{33}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 11497573.7472 \) (assuming GRH)
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

7.7.177885288000.1

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 R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.14.14.40$x^{14} + 2 x^{9} + 2 x^{4} + 2 x^{2} + 2 x + 2$$14$$1$$14$14T11$[8/7, 8/7, 8/7]_{7}^{3}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed