Properties

Label 12.0.1452563570950144.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{16}\cdot 53^{6}$
Root discriminant $18.34$
Ramified primes $2, 53$
Class number $4$
Class group $[4]$
Galois Group $S_4$ (as 12T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![54, -180, 426, -650, 777, -712, 551, -334, 172, -70, 25, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 172*x^8 - 334*x^7 + 551*x^6 - 712*x^5 + 777*x^4 - 650*x^3 + 426*x^2 - 180*x + 54)
gp: K = bnfinit(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 172*x^8 - 334*x^7 + 551*x^6 - 712*x^5 + 777*x^4 - 650*x^3 + 426*x^2 - 180*x + 54, 1)

Normalized defining polynomial

\(x^{12} \) \(\mathstrut -\mathstrut 6 x^{11} \) \(\mathstrut +\mathstrut 25 x^{10} \) \(\mathstrut -\mathstrut 70 x^{9} \) \(\mathstrut +\mathstrut 172 x^{8} \) \(\mathstrut -\mathstrut 334 x^{7} \) \(\mathstrut +\mathstrut 551 x^{6} \) \(\mathstrut -\mathstrut 712 x^{5} \) \(\mathstrut +\mathstrut 777 x^{4} \) \(\mathstrut -\mathstrut 650 x^{3} \) \(\mathstrut +\mathstrut 426 x^{2} \) \(\mathstrut -\mathstrut 180 x \) \(\mathstrut +\mathstrut 54 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 6]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1452563570950144=2^{16}\cdot 53^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $18.34$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{603} a^{10} - \frac{5}{603} a^{9} + \frac{26}{603} a^{8} - \frac{74}{603} a^{7} + \frac{254}{603} a^{6} + \frac{79}{603} a^{5} - \frac{86}{201} a^{4} + \frac{107}{603} a^{3} - \frac{61}{603} a^{2} - \frac{23}{201} a - \frac{1}{67}$, $\frac{1}{181503} a^{11} + \frac{145}{181503} a^{10} - \frac{19618}{181503} a^{9} + \frac{28348}{181503} a^{8} - \frac{5821}{181503} a^{7} - \frac{6290}{25929} a^{6} - \frac{25616}{60501} a^{5} - \frac{63517}{181503} a^{4} - \frac{1942}{4221} a^{3} + \frac{740}{20167} a^{2} + \frac{10282}{60501} a + \frac{2396}{20167}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C4, order $4$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $5$
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
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 357.3814526 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_4$ (as 12T9):

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

\(\Q(\sqrt{-53}) \), 3.1.212.1 x3, 6.0.38112512.1, 6.2.179776.1, 6.0.9528128.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$V_4$$[2]^{2}$
53Data not computed