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Properties

Label 12.0.85282689024.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 13^{4}$
Root discriminant $8.15$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $C_6\times S_3$ (as 12T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 4, -4, 5, -4, 6, -4, 1, -4, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + x^10 - 4*x^9 + x^8 - 4*x^7 + 6*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^12 + x^10 - 4*x^9 + x^8 - 4*x^7 + 6*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 4*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

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

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:  \(85282689024=2^{12}\cdot 3^{6}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8.15$
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:  $2, 3, 13$
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:  \( -a^{11} - a^{9} + 4 a^{8} - a^{7} + 3 a^{6} - 6 a^{5} + 3 a^{4} - 3 a^{3} + 4 a^{2} - 2 a + 1 \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a^{11} + a^{9} - 3 a^{8} + a^{7} - 3 a^{6} + 4 a^{5} - 3 a^{4} + 3 a^{3} - 2 a^{2} + 2 a - 1 \),  \( a^{11} + a^{10} + a^{9} - 3 a^{8} - 3 a^{7} - 3 a^{6} + 3 a^{5} + 2 a^{4} + 2 a^{3} - a^{2} \),  \( a^{10} + a^{8} - 4 a^{7} - 4 a^{5} + 5 a^{4} - a^{3} + 4 a^{2} - a + 1 \),  \( a^{10} + a^{9} + a^{8} - 2 a^{7} - 2 a^{6} - 2 a^{5} + a^{4} - a + 1 \),  \( a^{10} - 4 a^{7} - a^{5} + 5 a^{4} - a^{3} + a^{2} - a + 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11.0477502166 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 12T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.10816.1

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

Sibling fields

Galois closure: data not computed
Degree 18 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e2_3.2t1.1c1$1$ $ 2^{2} \cdot 3 $ $x^{2} - 3$ $C_2$ (as 2T1) $1$ $1$
1.3_13.6t1.1c1$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3_13.6t1.2c1$1$ $ 2^{2} \cdot 3 \cdot 13 $ $x^{6} - 2 x^{5} - 16 x^{4} + 18 x^{3} + 39 x^{2} - 52 x + 13$ $C_6$ (as 6T1) $0$ $1$
1.2e2_13.6t1.1c1$1$ $ 2^{2} \cdot 13 $ $x^{6} + 9 x^{4} + 14 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_13.6t1.1c2$1$ $ 2^{2} \cdot 13 $ $x^{6} + 9 x^{4} + 14 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3_13.6t1.1c2$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3_13.6t1.2c2$1$ $ 2^{2} \cdot 3 \cdot 13 $ $x^{6} - 2 x^{5} - 16 x^{4} + 18 x^{3} + 39 x^{2} - 52 x + 13$ $C_6$ (as 6T1) $0$ $1$
2.2e2_13e2.3t2.1c1$2$ $ 2^{2} \cdot 13^{2}$ $x^{3} - x^{2} - 4 x + 12$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3e2_13e2.6t3.1c1$2$ $ 2^{2} \cdot 3^{2} \cdot 13^{2}$ $x^{6} - 2 x^{5} - 8 x^{4} + 42 x^{3} - 108 x^{2} + 162 x - 99$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_13.6t5.1c1$2$ $ 2^{2} \cdot 13 $ $x^{6} - x^{4} - 2 x^{3} + 2 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3e2_13.12t18.2c1$2$ $ 2^{2} \cdot 3^{2} \cdot 13 $ $x^{12} + x^{10} - 4 x^{9} + x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + 5 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1$ $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.2e2_3e2_13.12t18.2c2$2$ $ 2^{2} \cdot 3^{2} \cdot 13 $ $x^{12} + x^{10} - 4 x^{9} + x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + 5 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1$ $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.2e2_13.6t5.1c2$2$ $ 2^{2} \cdot 13 $ $x^{6} - x^{4} - 2 x^{3} + 2 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.