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Properties

Label 14.6.16931894122...7632.1
Degree $14$
Signature $[6, 4]$
Discriminant $2^{26}\cdot 3^{12}\cdot 7^{15}$
Root discriminant $74.73$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T46

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-36288, 0, -9072, 0, -6048, 0, -3024, 0, 504, 0, 168, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 28*x^12 + 168*x^10 + 504*x^8 - 3024*x^6 - 6048*x^4 - 9072*x^2 - 36288)
 
gp: K = bnfinit(x^14 - 28*x^12 + 168*x^10 + 504*x^8 - 3024*x^6 - 6048*x^4 - 9072*x^2 - 36288, 1)
 

Normalized defining polynomial

\( x^{14} - 28 x^{12} + 168 x^{10} + 504 x^{8} - 3024 x^{6} - 6048 x^{4} - 9072 x^{2} - 36288 \)

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:  \(169318941227702858004037632=2^{26}\cdot 3^{12}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.73$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{6} - \frac{1}{12} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{7} - \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{144} a^{8} - \frac{1}{36} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{144} a^{9} + \frac{1}{72} a^{7} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{144} a^{10} - \frac{1}{36} a^{6} - \frac{1}{12} a^{4}$, $\frac{1}{864} a^{11} + \frac{1}{432} a^{9} + \frac{1}{72} a^{7} - \frac{1}{24} a^{5} - \frac{1}{12} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{5921856} a^{12} + \frac{373}{2960928} a^{10} - \frac{1}{288} a^{9} + \frac{17}{54832} a^{8} + \frac{1}{72} a^{7} + \frac{12227}{493488} a^{6} - \frac{1}{24} a^{5} + \frac{7675}{82248} a^{4} - \frac{5}{24} a^{3} + \frac{1628}{3427} a^{2} - \frac{1}{2} a + \frac{1501}{3427}$, $\frac{1}{5921856} a^{13} + \frac{373}{2960928} a^{11} - \frac{1}{288} a^{10} + \frac{17}{54832} a^{9} - \frac{8335}{493488} a^{7} + \frac{1}{72} a^{6} + \frac{821}{82248} a^{5} + \frac{1}{24} a^{4} - \frac{171}{6854} a^{3} + \frac{1501}{3427} a$

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:  $9$
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:  \( 554978941.697 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T46:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for 2[1/2]S(7)
Character table for 2[1/2]S(7)

Intermediate fields

\(\Q(\sqrt{7}) \), 7.3.614771555328.1

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

Sibling fields

Degree 7 sibling: 7.3.614771555328.1
Degree 21 sibling: Deg 21
Degree 30 sibling: data not computed
Degree 35 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.12.24.392$x^{12} + 2 x^{10} + 4 x^{9} + 4 x^{8} + 4 x^{7} - 2 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 4 x + 2$$12$$1$$24$$C_2 \times S_4$$[2, 8/3, 8/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
7Data not computed