Properties

Label 12.0.1339147769319424.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 83^{6}$
Root discriminant $18.22$
Ramified primes $2, 83$
Class number $4$
Class group $[4]$
Galois Group $C_2 \times S_4$ (as 12T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![526, 1290, 771, -394, -287, 222, 22, -106, 27, 24, -10, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 10*x^10 + 24*x^9 + 27*x^8 - 106*x^7 + 22*x^6 + 222*x^5 - 287*x^4 - 394*x^3 + 771*x^2 + 1290*x + 526)
gp: K = bnfinit(x^12 - 2*x^11 - 10*x^10 + 24*x^9 + 27*x^8 - 106*x^7 + 22*x^6 + 222*x^5 - 287*x^4 - 394*x^3 + 771*x^2 + 1290*x + 526, 1)

Normalized defining polynomial

\(x^{12} \) \(\mathstrut -\mathstrut 2 x^{11} \) \(\mathstrut -\mathstrut 10 x^{10} \) \(\mathstrut +\mathstrut 24 x^{9} \) \(\mathstrut +\mathstrut 27 x^{8} \) \(\mathstrut -\mathstrut 106 x^{7} \) \(\mathstrut +\mathstrut 22 x^{6} \) \(\mathstrut +\mathstrut 222 x^{5} \) \(\mathstrut -\mathstrut 287 x^{4} \) \(\mathstrut -\mathstrut 394 x^{3} \) \(\mathstrut +\mathstrut 771 x^{2} \) \(\mathstrut +\mathstrut 1290 x \) \(\mathstrut +\mathstrut 526 \)

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:  \(1339147769319424=2^{12}\cdot 83^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $18.22$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2988158496580} a^{11} + \frac{98604251073}{597631699316} a^{10} - \frac{106779345455}{597631699316} a^{9} - \frac{130004083101}{2988158496580} a^{8} - \frac{183809811}{3474602903} a^{7} + \frac{529826155447}{1494079248290} a^{6} - \frac{35282236057}{149407924829} a^{5} + \frac{568910268941}{1494079248290} a^{4} + \frac{14188142829}{69492058060} a^{3} + \frac{283907429599}{597631699316} a^{2} + \frac{335357640984}{747039624145} a - \frac{447393549649}{1494079248290}$

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

Class group and class number

$C_{4}$, which has 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:  \( 265.254382815 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times S_4$ (as 12T24):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 48
The 10 conjugacy class representatives for $C_2 \times S_4$
Character table for $C_2 \times S_4$

Intermediate fields

\(\Q(\sqrt{83}) \), 3.1.83.1, 6.2.36594368.1, 6.0.27556.1, 6.0.9148592.1

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

Sibling fields

Degree 6 siblings: data not computed
Degree 8 siblings: data not computed
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$83$83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$