Properties

Label 12.2.27406859829248.2
Degree $12$
Signature $[2, 5]$
Discriminant $-\,2^{16}\cdot 53^{5}$
Root discriminant $13.18$
Ramified primes $2, 53$
Class number $1$
Class group Trivial
Galois Group $S_4$ (as 12T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -18, 24, -32, 34, -32, 24, -18, 10, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 18*x^9 + 24*x^8 - 32*x^7 + 34*x^6 - 32*x^5 + 24*x^4 - 18*x^3 + 10*x^2 - 4*x + 1)
gp: K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 18*x^9 + 24*x^8 - 32*x^7 + 34*x^6 - 32*x^5 + 24*x^4 - 18*x^3 + 10*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{12} \) \(\mathstrut -\mathstrut 4 x^{11} \) \(\mathstrut +\mathstrut 10 x^{10} \) \(\mathstrut -\mathstrut 18 x^{9} \) \(\mathstrut +\mathstrut 24 x^{8} \) \(\mathstrut -\mathstrut 32 x^{7} \) \(\mathstrut +\mathstrut 34 x^{6} \) \(\mathstrut -\mathstrut 32 x^{5} \) \(\mathstrut +\mathstrut 24 x^{4} \) \(\mathstrut -\mathstrut 18 x^{3} \) \(\mathstrut +\mathstrut 10 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 5]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-27406859829248=-\,2^{16}\cdot 53^{5}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.18$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 53$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{1}{2} a^{11} - 3 a^{10} + \frac{19}{2} a^{9} - 20 a^{8} + \frac{61}{2} a^{7} - 37 a^{6} + \frac{77}{2} a^{5} - 36 a^{4} + \frac{51}{2} a^{3} - 15 a^{2} + \frac{13}{2} a - 2 \),  \( a^{11} - 4 a^{10} + 10 a^{9} - 18 a^{8} + 24 a^{7} - 32 a^{6} + 34 a^{5} - 32 a^{4} + 24 a^{3} - 18 a^{2} + 10 a - 4 \),  \( \frac{3}{2} a^{11} - 7 a^{10} + \frac{37}{2} a^{9} - 35 a^{8} + \frac{97}{2} a^{7} - 61 a^{6} + \frac{133}{2} a^{5} - 59 a^{4} + \frac{89}{2} a^{3} - 29 a^{2} + \frac{33}{2} a - 5 \),  \( \frac{1}{2} a^{11} - 3 a^{10} + \frac{17}{2} a^{9} - 17 a^{8} + \frac{49}{2} a^{7} - 29 a^{6} + \frac{65}{2} a^{5} - 27 a^{4} + \frac{41}{2} a^{3} - 11 a^{2} + \frac{11}{2} a - 1 \),  \( \frac{3}{2} a^{11} - \frac{11}{2} a^{10} + \frac{25}{2} a^{9} - \frac{39}{2} a^{8} + \frac{41}{2} a^{7} - \frac{49}{2} a^{6} + \frac{43}{2} a^{5} - \frac{37}{2} a^{4} + \frac{19}{2} a^{3} - \frac{17}{2} a^{2} + \frac{7}{2} a - \frac{3}{2} \),  \( \frac{5}{2} a^{11} - \frac{21}{2} a^{10} + \frac{51}{2} a^{9} - \frac{87}{2} a^{8} + \frac{105}{2} a^{7} - \frac{125}{2} a^{6} + \frac{127}{2} a^{5} - \frac{105}{2} a^{4} + \frac{65}{2} a^{3} - \frac{43}{2} a^{2} + \frac{25}{2} a - \frac{7}{2} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 87.2121552468 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_4$ (as 12T8):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

3.1.212.1, 4.2.848.1 x2, 6.2.179776.1

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

Sibling fields

Galois closure: data not computed
Degree 4 sibling: data not computed
Degree 6 siblings: data not computed
Degree 8 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$53$53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$