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Properties

Label 12.0.72964670628096.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 3^{10}\cdot 13^{6}$
Root discriminant $14.30$
Ramified primes $2, 3, 13$
Class number $2$
Class group $[2]$
Galois group $S_3^2$ (as 12T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 23, 24, 37, -35, 52, -5, -5, 12, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - x^10 + 12*x^9 - 5*x^8 - 5*x^7 + 52*x^6 - 35*x^5 + 37*x^4 + 24*x^3 + 23*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^12 - x^11 - x^10 + 12*x^9 - 5*x^8 - 5*x^7 + 52*x^6 - 35*x^5 + 37*x^4 + 24*x^3 + 23*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - x^{10} + 12 x^{9} - 5 x^{8} - 5 x^{7} + 52 x^{6} - 35 x^{5} + 37 x^{4} + 24 x^{3} + 23 x^{2} + 5 x + 1 \)

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:  \(72964670628096=2^{8}\cdot 3^{10}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.30$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{4} - \frac{4}{9} a^{2} - \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{6} - \frac{1}{18} a^{3} + \frac{1}{18}$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{7} - \frac{1}{18} a^{4} + \frac{1}{18} a$, $\frac{1}{594} a^{11} - \frac{1}{198} a^{10} + \frac{5}{594} a^{9} - \frac{31}{594} a^{8} - \frac{25}{198} a^{7} + \frac{13}{594} a^{6} - \frac{73}{594} a^{5} + \frac{37}{198} a^{4} + \frac{211}{594} a^{3} + \frac{229}{594} a^{2} + \frac{31}{198} a - \frac{49}{594}$

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:  \( -\frac{13}{66} a^{11} + \frac{20}{99} a^{10} + \frac{2}{11} a^{9} - \frac{463}{198} a^{8} + \frac{104}{99} a^{7} + \frac{82}{99} a^{6} - \frac{219}{22} a^{5} + \frac{712}{99} a^{4} - \frac{266}{33} a^{3} - \frac{593}{198} a^{2} - \frac{488}{99} a - \frac{7}{99} \) (order $6$)
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:  \( 530.218430458 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 12T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{13})\), 6.2.2847312.1 x3

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 6 sibling: data not computed
Degree 9 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$