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Properties

Label 12.6.53804562284000000.1
Degree $12$
Signature $[6, 3]$
Discriminant $-\,2^{8}\cdot 5^{6}\cdot 11^{5}\cdot 17^{4}$
Root discriminant $24.79$
Ramified primes $2, 5, 11, 17$
Class number $1$
Class group Trivial
Galois group $D_6:D_6$ (as 12T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1061, 23, 1131, -347, -303, 42, 139, 12, -33, 17, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 8*x^10 + 17*x^9 - 33*x^8 + 12*x^7 + 139*x^6 + 42*x^5 - 303*x^4 - 347*x^3 + 1131*x^2 + 23*x - 1061)
 
gp: K = bnfinit(x^12 - 2*x^11 - 8*x^10 + 17*x^9 - 33*x^8 + 12*x^7 + 139*x^6 + 42*x^5 - 303*x^4 - 347*x^3 + 1131*x^2 + 23*x - 1061, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 8 x^{10} + 17 x^{9} - 33 x^{8} + 12 x^{7} + 139 x^{6} + 42 x^{5} - 303 x^{4} - 347 x^{3} + 1131 x^{2} + 23 x - 1061 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-53804562284000000=-\,2^{8}\cdot 5^{6}\cdot 11^{5}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.79$
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:  $2, 5, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{8} - \frac{1}{15} a^{7} + \frac{7}{15} a^{5} + \frac{4}{15} a^{4} - \frac{1}{3} a^{3} - \frac{2}{15} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{9} - \frac{1}{15} a^{7} + \frac{2}{15} a^{6} + \frac{1}{15} a^{5} - \frac{1}{15} a^{4} + \frac{1}{5} a^{3} + \frac{2}{15} a^{2} - \frac{2}{15} a + \frac{4}{15}$, $\frac{1}{15} a^{10} + \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{2}{5} a^{5} + \frac{7}{15} a^{4} - \frac{1}{5} a^{3} - \frac{4}{15} a^{2} - \frac{2}{15} a - \frac{2}{5}$, $\frac{1}{8684727037635} a^{11} + \frac{172245812912}{8684727037635} a^{10} - \frac{160253929192}{8684727037635} a^{9} - \frac{3286736855}{157904127957} a^{8} - \frac{39731791753}{263173546595} a^{7} + \frac{1048428138319}{8684727037635} a^{6} - \frac{108467573284}{578981802509} a^{5} + \frac{3839431159424}{8684727037635} a^{4} - \frac{14126218481}{280152485085} a^{3} - \frac{168132451297}{8684727037635} a^{2} - \frac{23886247026}{578981802509} a - \frac{2864955084634}{8684727037635}$

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:  $8$
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:  \( 6053.57336949 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6:D_6$ (as 12T81):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $D_6:D_6$
Character table for $D_6:D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 6.6.69938000.1

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 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/13.12.0.1}{12} }$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.8.2$x^{12} - 8 x^{3} + 16$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.5.1$x^{6} - 11$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$