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Properties

Label 12.0.101559956668416.14
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}$
Root discriminant $14.70$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_6\times C_2$ (as 12T219)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 18, 27, 30, 21, -6, -2, -6, 3, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^9 + 3*x^8 - 6*x^7 - 2*x^6 - 6*x^5 + 21*x^4 + 30*x^3 + 27*x^2 + 18*x + 9)
 
gp: K = bnfinit(x^12 - 4*x^9 + 3*x^8 - 6*x^7 - 2*x^6 - 6*x^5 + 21*x^4 + 30*x^3 + 27*x^2 + 18*x + 9, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{9} + 3 x^{8} - 6 x^{7} - 2 x^{6} - 6 x^{5} + 21 x^{4} + 30 x^{3} + 27 x^{2} + 18 x + 9 \)

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:  \(101559956668416=2^{18}\cdot 3^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.70$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{801} a^{11} - \frac{2}{801} a^{10} + \frac{4}{801} a^{9} - \frac{4}{267} a^{8} + \frac{116}{801} a^{7} + \frac{118}{801} a^{6} + \frac{29}{801} a^{5} - \frac{242}{801} a^{4} + \frac{109}{267} a^{3} - \frac{10}{89} a^{2} - \frac{109}{267} a - \frac{44}{89}$

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:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{46}{801} a^{11} + \frac{1}{267} a^{10} - \frac{2}{267} a^{9} + \frac{196}{801} a^{8} - \frac{58}{267} a^{7} + \frac{119}{267} a^{6} - \frac{88}{801} a^{5} + \frac{121}{267} a^{4} - \frac{386}{267} a^{3} - \frac{311}{267} a^{2} - \frac{168}{89} a - \frac{23}{89} \) (order $6$)
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:  \( 635.385098052 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_6\times C_2$ (as 12T219):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 22 conjugacy class representatives for $S_6\times C_2$
Character table for $S_6\times C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 6.2.3359232.3

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

Sibling fields

Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.26$x^{12} - 12 x^{10} - 2340 x^{8} + 1120 x^{6} + 1264 x^{4} - 192 x^{2} + 3648$$2$$6$$18$$C_2^2 \times A_4$$[2, 2, 3]^{6}$
$3$3.6.11.7$x^{6} + 21$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$