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Properties

Label 12.0.8226356490141696.17
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{22}$
Root discriminant $21.20$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $D_6$ (as 12T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -180, 372, 432, 186, 72, -50, -72, 24, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 + 24*x^8 - 72*x^7 - 50*x^6 + 72*x^5 + 186*x^4 + 432*x^3 + 372*x^2 - 180*x + 25)
 
gp: K = bnfinit(x^12 - 6*x^10 + 24*x^8 - 72*x^7 - 50*x^6 + 72*x^5 + 186*x^4 + 432*x^3 + 372*x^2 - 180*x + 25, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{10} + 24 x^{8} - 72 x^{7} - 50 x^{6} + 72 x^{5} + 186 x^{4} + 432 x^{3} + 372 x^{2} - 180 x + 25 \)

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:  \(8226356490141696=2^{18}\cdot 3^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.20$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{432890} a^{10} + \frac{78609}{432890} a^{9} + \frac{7985}{43289} a^{8} - \frac{10303}{43289} a^{7} + \frac{22682}{216445} a^{6} + \frac{20637}{216445} a^{5} + \frac{122191}{432890} a^{4} + \frac{37391}{432890} a^{3} - \frac{20143}{43289} a^{2} - \frac{75669}{216445} a + \frac{11708}{43289}$, $\frac{1}{169259990} a^{11} - \frac{29}{169259990} a^{10} - \frac{16835456}{84629995} a^{9} + \frac{3089020}{16925999} a^{8} + \frac{1785103}{7359130} a^{7} + \frac{41494527}{169259990} a^{6} + \frac{67757439}{169259990} a^{5} - \frac{77013557}{169259990} a^{4} + \frac{10206356}{84629995} a^{3} - \frac{32022354}{84629995} a^{2} - \frac{6254061}{169259990} a - \frac{16712769}{33851998}$

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:  \( -\frac{1848}{142715} a^{11} - \frac{366}{142715} a^{10} + \frac{10754}{142715} a^{9} + \frac{921}{57086} a^{8} - \frac{1899}{6205} a^{7} + \frac{122292}{142715} a^{6} + \frac{115371}{142715} a^{5} - \frac{87672}{142715} a^{4} - \frac{387314}{142715} a^{3} - \frac{1684167}{285430} a^{2} - \frac{811473}{142715} a + \frac{48071}{28543} \) (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:  \( 9382.11631752 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6$ (as 12T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), 3.1.243.1 x3, \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.0.177147.2, 6.2.90699264.1 x3, 6.0.30233088.2 x3

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

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$