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Properties

Label 12.0.6499837226778624.37
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{18}$
Root discriminant $20.78$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
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![33, 36, 36, 96, 57, -108, -104, 36, 54, -4, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 4*x^9 + 54*x^8 + 36*x^7 - 104*x^6 - 108*x^5 + 57*x^4 + 96*x^3 + 36*x^2 + 36*x + 33)
 
gp: K = bnfinit(x^12 - 12*x^10 - 4*x^9 + 54*x^8 + 36*x^7 - 104*x^6 - 108*x^5 + 57*x^4 + 96*x^3 + 36*x^2 + 36*x + 33, 1)
 

Normalized defining polynomial

\( x^{12} - 12 x^{10} - 4 x^{9} + 54 x^{8} + 36 x^{7} - 104 x^{6} - 108 x^{5} + 57 x^{4} + 96 x^{3} + 36 x^{2} + 36 x + 33 \)

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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{21} a^{9} - \frac{3}{7} a^{7} + \frac{1}{3} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{3} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{21} a^{10} - \frac{3}{7} a^{8} + \frac{1}{3} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{4} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{693} a^{11} - \frac{10}{693} a^{10} - \frac{1}{63} a^{9} + \frac{139}{693} a^{8} + \frac{248}{693} a^{7} + \frac{31}{693} a^{6} - \frac{4}{33} a^{5} + \frac{79}{231} a^{4} - \frac{4}{77} a^{3} + \frac{53}{231} a^{2} + \frac{109}{231} a + \frac{4}{21}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 951.410419675 \)
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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{3})\), 6.0.10077696.3

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.6.9.6$x^{6} + 3 x^{4} + 6 x^{3} + 6$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
3.6.9.6$x^{6} + 3 x^{4} + 6 x^{3} + 6$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_3.2t1.1c1$1$ $ 2^{2} \cdot 3 $ $x^{2} - 3$ $C_2$ (as 2T1) $1$ $1$
* 1.2e3_3.2t1.2c1$1$ $ 2^{3} \cdot 3 $ $x^{2} + 6$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e3.2t1.2c1$1$ $ 2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_3e2.6t1.3c1$1$ $ 2^{3} \cdot 3^{2}$ $x^{6} + 12 x^{4} + 36 x^{2} + 8$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e2.6t1.1c1$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} - 6 x^{4} + 9 x^{2} - 3$ $C_6$ (as 6T1) $0$ $1$
1.2e2_3e2.6t1.1c2$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} - 6 x^{4} + 9 x^{2} - 3$ $C_6$ (as 6T1) $0$ $1$
1.2e3_3e2.6t1.4c1$1$ $ 2^{3} \cdot 3^{2}$ $x^{6} + 12 x^{4} + 36 x^{2} + 24$ $C_6$ (as 6T1) $0$ $-1$
1.2e3_3e2.6t1.3c2$1$ $ 2^{3} \cdot 3^{2}$ $x^{6} + 12 x^{4} + 36 x^{2} + 8$ $C_6$ (as 6T1) $0$ $-1$
1.2e3_3e2.6t1.4c2$1$ $ 2^{3} \cdot 3^{2}$ $x^{6} + 12 x^{4} + 36 x^{2} + 24$ $C_6$ (as 6T1) $0$ $-1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
2.2e3_3e3.3t2.1c1$2$ $ 2^{3} \cdot 3^{3}$ $x^{3} + 3 x - 2$ $S_3$ (as 3T2) $1$ $0$
2.2e5_3e3.6t3.4c1$2$ $ 2^{5} \cdot 3^{3}$ $x^{6} - 3 x^{4} - 8 x^{3} - 3 x^{2} + 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e3_3e4.6t5.1c1$2$ $ 2^{3} \cdot 3^{4}$ $x^{6} + 6 x^{4} - 2 x^{3} + 9 x^{2} - 6 x + 7$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e3_3e4.6t5.1c2$2$ $ 2^{3} \cdot 3^{4}$ $x^{6} + 6 x^{4} - 2 x^{3} + 9 x^{2} - 6 x + 7$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e5_3e4.12t18.3c1$2$ $ 2^{5} \cdot 3^{4}$ $x^{12} - 12 x^{10} - 4 x^{9} + 54 x^{8} + 36 x^{7} - 104 x^{6} - 108 x^{5} + 57 x^{4} + 96 x^{3} + 36 x^{2} + 36 x + 33$ $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.2e5_3e4.12t18.3c2$2$ $ 2^{5} \cdot 3^{4}$ $x^{12} - 12 x^{10} - 4 x^{9} + 54 x^{8} + 36 x^{7} - 104 x^{6} - 108 x^{5} + 57 x^{4} + 96 x^{3} + 36 x^{2} + 36 x + 33$ $C_6\times S_3$ (as 12T18) $0$ $0$

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 *.