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Properties

Label 12.0.81462863889.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 11^{2}\cdot 31^{4}$
Root discriminant $8.11$
Ramified primes $3, 11, 31$
Class number $1$
Class group Trivial
Galois group $C_2^2\times S_4$ (as 12T48)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\( x^{12} - x^{11} + 2 x^{10} - 3 x^{9} + 3 x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 3 x^{4} - 4 x^{3} + x^{2} - x + 1 \)

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:  \(81462863889=3^{6}\cdot 11^{2}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8.11$
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:  $3, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$

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:  \( -2 a^{11} + a^{10} - 5 a^{9} + 5 a^{8} - 6 a^{7} + 6 a^{6} - 10 a^{5} + 3 a^{4} - 9 a^{3} + 5 a^{2} - a + 4 \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( 2 a^{11} - a^{10} + 4 a^{9} - 4 a^{8} + 5 a^{7} - 5 a^{6} + 9 a^{5} - 3 a^{4} + 7 a^{3} - 5 a^{2} + a - 3 \),  \( a^{11} + a^{10} + a^{9} + 3 a^{5} + 3 a^{4} + 3 a^{3} - a^{2} - 2 a - 2 \),  \( 2 a^{11} - a^{10} + 3 a^{9} - 3 a^{8} + 3 a^{7} - 2 a^{6} + 6 a^{5} + 2 a^{3} - 2 a^{2} - 2 a \),  \( 2 a^{11} + 4 a^{9} - 3 a^{8} + 4 a^{7} - 4 a^{6} + 9 a^{5} + 8 a^{3} - 4 a^{2} - 3 \),  \( a^{2} + 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5.40994597563 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times S_4$ (as 12T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 20 conjugacy class representatives for $C_2^2\times S_4$
Character table for $C_2^2\times S_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.31.1, 6.0.25947.1, 6.0.10571.1, 6.2.285417.1

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 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$