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Properties

Label 14.0.20843694053...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 5^{7}\cdot 7^{18}$
Root discriminant $54.58$
Ramified primes $2, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 14T25

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56, -168, -28, 560, -994, 742, 1225, -2536, 2422, -896, 525, -42, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 42*x^12 - 42*x^11 + 525*x^10 - 896*x^9 + 2422*x^8 - 2536*x^7 + 1225*x^6 + 742*x^5 - 994*x^4 + 560*x^3 - 28*x^2 - 168*x + 56)
 
gp: K = bnfinit(x^14 + 42*x^12 - 42*x^11 + 525*x^10 - 896*x^9 + 2422*x^8 - 2536*x^7 + 1225*x^6 + 742*x^5 - 994*x^4 + 560*x^3 - 28*x^2 - 168*x + 56, 1)
 

Normalized defining polynomial

\( x^{14} + 42 x^{12} - 42 x^{11} + 525 x^{10} - 896 x^{9} + 2422 x^{8} - 2536 x^{7} + 1225 x^{6} + 742 x^{5} - 994 x^{4} + 560 x^{3} - 28 x^{2} - 168 x + 56 \)

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:  \(-2084369405325374720000000=-\,2^{14}\cdot 5^{7}\cdot 7^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.58$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{30} a^{9} - \frac{1}{15} a^{8} + \frac{2}{15} a^{7} + \frac{7}{30} a^{6} + \frac{2}{5} a^{5} + \frac{1}{30} a^{4} + \frac{1}{30} a^{3} + \frac{13}{30} a^{2} + \frac{1}{5} a + \frac{4}{15}$, $\frac{1}{60} a^{10} + \frac{11}{60} a^{6} + \frac{1}{6} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a^{3} + \frac{17}{60} a^{2} - \frac{1}{6} a - \frac{7}{30}$, $\frac{1}{120} a^{11} - \frac{1}{60} a^{9} - \frac{1}{20} a^{8} + \frac{13}{120} a^{7} - \frac{1}{30} a^{6} - \frac{23}{60} a^{5} - \frac{1}{10} a^{4} - \frac{7}{24} a^{3} + \frac{9}{20} a^{2} + \frac{9}{20} a - \frac{7}{15}$, $\frac{1}{25920} a^{12} + \frac{41}{12960} a^{11} - \frac{1}{360} a^{10} + \frac{61}{12960} a^{9} + \frac{17}{25920} a^{8} + \frac{607}{2592} a^{7} + \frac{77}{432} a^{6} - \frac{457}{1296} a^{5} - \frac{2887}{25920} a^{4} + \frac{163}{810} a^{3} + \frac{1471}{3240} a^{2} - \frac{263}{810} a - \frac{2779}{6480}$, $\frac{1}{3862416960} a^{13} + \frac{16639}{1931208480} a^{12} + \frac{314197}{241401060} a^{11} - \frac{14778587}{1931208480} a^{10} - \frac{39156983}{3862416960} a^{9} + \frac{998033}{23842080} a^{8} + \frac{196268101}{965604240} a^{7} - \frac{61938797}{965604240} a^{6} + \frac{27505709}{85831488} a^{5} + \frac{64600535}{193120848} a^{4} + \frac{144797987}{482802120} a^{3} - \frac{29690357}{241401060} a^{2} - \frac{294794963}{965604240} a - \frac{80905831}{241401060}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -1 \) (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:  \( 15378453.4927 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T25:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 19 conjugacy class representatives for [7^2:6_3]2
Character table for [7^2:6_3]2

Intermediate fields

\(\Q(\sqrt{-5}) \)

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

Sibling fields

Degree 21 siblings: data not computed
Degree 28 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.7.11.3$x^{7} + 14 x^{5} + 7$$7$$1$$11$$F_7$$[11/6]_{6}$
7.7.7.3$x^{7} + 35 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$