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Properties

Label 12.0.458793060873472.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 13^{11}$
Root discriminant $16.66$
Ramified primes $2, 13$
Class number $3$
Class group $[3]$
Galois group $S_3 \times C_4$ (as 12T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, -9, -36, -40, 152, -185, 157, -100, 55, -27, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 9*x^10 - 27*x^9 + 55*x^8 - 100*x^7 + 157*x^6 - 185*x^5 + 152*x^4 - 40*x^3 - 36*x^2 - 9*x + 27)
 
gp: K = bnfinit(x^12 - 3*x^11 + 9*x^10 - 27*x^9 + 55*x^8 - 100*x^7 + 157*x^6 - 185*x^5 + 152*x^4 - 40*x^3 - 36*x^2 - 9*x + 27, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 9 x^{10} - 27 x^{9} + 55 x^{8} - 100 x^{7} + 157 x^{6} - 185 x^{5} + 152 x^{4} - 40 x^{3} - 36 x^{2} - 9 x + 27 \)

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:  \(458793060873472=2^{8}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.66$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{4}{27} a^{7} + \frac{1}{9} a^{6} + \frac{1}{27} a^{4} - \frac{1}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{81} a^{11} - \frac{1}{81} a^{10} - \frac{2}{81} a^{9} - \frac{4}{81} a^{8} + \frac{2}{81} a^{7} - \frac{2}{27} a^{6} - \frac{26}{81} a^{5} + \frac{11}{27} a^{4} - \frac{7}{81} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{1}{3}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 295.950151185 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_3$ (as 12T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 12 conjugacy class representatives for $S_3 \times C_4$
Character table for $S_3 \times C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.1.676.1, 4.0.2197.1, 6.2.5940688.1

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 12 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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$13$13.12.11.2$x^{12} - 52$$12$$1$$11$$C_{12}$$[\ ]_{12}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
1.2e2_13.2t1.1c1$1$ $ 2^{2} \cdot 13 $ $x^{2} + 13$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_13.4t1.1c1$1$ $ 2^{2} \cdot 13 $ $x^{4} - 13 x^{2} + 13$ $C_4$ (as 4T1) $0$ $1$
* 1.13.4t1.1c1$1$ $ 13 $ $x^{4} - x^{3} + 2 x^{2} + 4 x + 3$ $C_4$ (as 4T1) $0$ $-1$
1.2e2_13.4t1.1c2$1$ $ 2^{2} \cdot 13 $ $x^{4} - 13 x^{2} + 13$ $C_4$ (as 4T1) $0$ $1$
* 1.13.4t1.1c2$1$ $ 13 $ $x^{4} - x^{3} + 2 x^{2} + 4 x + 3$ $C_4$ (as 4T1) $0$ $-1$
* 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_13e2.6t3.2c1$2$ $ 2^{2} \cdot 13^{2}$ $x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 16 x + 12$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_13e2.12t11.1c1$2$ $ 2^{2} \cdot 13^{2}$ $x^{12} - 3 x^{11} + 9 x^{10} - 27 x^{9} + 55 x^{8} - 100 x^{7} + 157 x^{6} - 185 x^{5} + 152 x^{4} - 40 x^{3} - 36 x^{2} - 9 x + 27$ $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.2e2_13e2.12t11.1c2$2$ $ 2^{2} \cdot 13^{2}$ $x^{12} - 3 x^{11} + 9 x^{10} - 27 x^{9} + 55 x^{8} - 100 x^{7} + 157 x^{6} - 185 x^{5} + 152 x^{4} - 40 x^{3} - 36 x^{2} - 9 x + 27$ $S_3 \times C_4$ (as 12T11) $0$ $0$

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 *.