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Properties

Label 12.12.1345081186...3104.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{20}\cdot 3^{12}\cdot 17^{6}$
Root discriminant $39.27$
Ramified primes $2, 3, 17$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 12T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-508, -1272, 108, 2372, 1260, -1080, -861, 180, 216, -10, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 24*x^10 - 10*x^9 + 216*x^8 + 180*x^7 - 861*x^6 - 1080*x^5 + 1260*x^4 + 2372*x^3 + 108*x^2 - 1272*x - 508)
 
gp: K = bnfinit(x^12 - 24*x^10 - 10*x^9 + 216*x^8 + 180*x^7 - 861*x^6 - 1080*x^5 + 1260*x^4 + 2372*x^3 + 108*x^2 - 1272*x - 508, 1)
 

Normalized defining polynomial

\( x^{12} - 24 x^{10} - 10 x^{9} + 216 x^{8} + 180 x^{7} - 861 x^{6} - 1080 x^{5} + 1260 x^{4} + 2372 x^{3} + 108 x^{2} - 1272 x - 508 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13450811861294383104=2^{20}\cdot 3^{12}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.27$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{12} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$

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

Class group and class number

Trivial group, which has 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:  $11$
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:  \( 1816144.09239 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 12T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 6.6.114610464.1

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

Sibling fields

Degree 6 siblings: 6.6.114610464.1, 6.6.25380864.1
Degree 9 sibling: 9.9.21389063233536.1
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 18 siblings: 18.18.287700265443621620596836604575744.1, Deg 18, Deg 18
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$3$3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$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$$[3/2, 3/2]_{2}^{2}$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$