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Properties

Label 12.0.6499837226778624.48
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{18}$
Root discriminant $20.78$
Ramified primes $2, 3$
Class number $6$
Class group $[6]$
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![262, -108, 198, -268, 225, -180, 102, -36, 36, -16, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 16*x^9 + 36*x^8 - 36*x^7 + 102*x^6 - 180*x^5 + 225*x^4 - 268*x^3 + 198*x^2 - 108*x + 262)
 
gp: K = bnfinit(x^12 - 16*x^9 + 36*x^8 - 36*x^7 + 102*x^6 - 180*x^5 + 225*x^4 - 268*x^3 + 198*x^2 - 108*x + 262, 1)
 

Normalized defining polynomial

\( x^{12} - 16 x^{9} + 36 x^{8} - 36 x^{7} + 102 x^{6} - 180 x^{5} + 225 x^{4} - 268 x^{3} + 198 x^{2} - 108 x + 262 \)

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:  \(6499837226778624=2^{24}\cdot 3^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.78$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{9} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} - \frac{2}{9}$, $\frac{1}{360} a^{10} - \frac{1}{45} a^{9} - \frac{1}{40} a^{8} - \frac{3}{40} a^{6} - \frac{1}{2} a^{5} - \frac{29}{120} a^{4} - \frac{1}{15} a^{3} - \frac{1}{5} a^{2} + \frac{7}{18} a + \frac{7}{180}$, $\frac{1}{119579760} a^{11} + \frac{6281}{23915952} a^{10} - \frac{1255493}{119579760} a^{9} - \frac{1989379}{39859920} a^{8} - \frac{2998369}{39859920} a^{7} - \frac{1360777}{39859920} a^{6} + \frac{2307577}{13286640} a^{5} + \frac{824221}{2657328} a^{4} - \frac{550537}{1660830} a^{3} - \frac{10721389}{29894940} a^{2} - \frac{20940503}{59789880} a + \frac{2358331}{59789880}$

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

Class group and class number

$C_{6}$, which has order $6$

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:  \( 1487.44608428 \)
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), 3.1.648.1 x3, \(\Q(\sqrt{-2}, \sqrt{3})\), 6.0.3359232.4, 6.0.40310784.2 x3, 6.2.20155392.5 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
$3$3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$