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Properties

Label 14.0.44311752545...4208.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 7^{9}\cdot 173^{6}$
Root discriminant $57.61$
Ramified primes $2, 7, 173$
Class number $216$ (GRH)
Class group $[216]$ (GRH)
Galois group 14T48

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 0, 727, 0, 3555, 0, 3843, 0, 1717, 0, 357, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 33*x^12 + 357*x^10 + 1717*x^8 + 3843*x^6 + 3555*x^4 + 727*x^2 + 7)
 
gp: K = bnfinit(x^14 + 33*x^12 + 357*x^10 + 1717*x^8 + 3843*x^6 + 3555*x^4 + 727*x^2 + 7, 1)
 

Normalized defining polynomial

\( x^{14} + 33 x^{12} + 357 x^{10} + 1717 x^{8} + 3843 x^{6} + 3555 x^{4} + 727 x^{2} + 7 \)

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:  \(-4431175254520053739614208=-\,2^{12}\cdot 7^{9}\cdot 173^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.61$
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:  $2, 7, 173$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{1}{32} a^{3} - \frac{1}{32} a$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{8} - \frac{5}{64} a^{4} + \frac{3}{64}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{3}{128} a^{9} + \frac{1}{128} a^{8} + \frac{1}{32} a^{7} + \frac{1}{32} a^{6} + \frac{7}{128} a^{5} - \frac{15}{128} a^{4} - \frac{1}{64} a^{3} - \frac{1}{64} a^{2} + \frac{53}{128} a - \frac{49}{128}$

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

Class group and class number

$C_{216}$, which has order $216$ (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:  \( 93324.5652468 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T48:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5376
The 40 conjugacy class representatives for [2^7]F_42(7)=2wrF_42(7)
Character table for [2^7]F_42(7)=2wrF_42(7) is not computed

Intermediate fields

7.7.12431698517.1

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 28 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$0$Trivial$[\ ]$
2.12.12.4$x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 38 x^{2} - 31$$2$$6$$12$12T87$[2, 2, 2, 2, 2]^{6}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$173$173.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
173.6.3.1$x^{6} - 346 x^{4} + 29929 x^{2} - 129442925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
173.6.3.1$x^{6} - 346 x^{4} + 29929 x^{2} - 129442925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$