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Properties

Label 12.0.342648531949824.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 3^{14}\cdot 23^{4}$
Root discriminant $16.26$
Ramified primes $2, 3, 23$
Class number $1$
Class group Trivial
Galois group 12T183

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 21, -34, 55, -13, 62, -25, 19, -10, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 5*x^10 - 10*x^9 + 19*x^8 - 25*x^7 + 62*x^6 - 13*x^5 + 55*x^4 - 34*x^3 + 21*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^12 - x^11 + 5*x^10 - 10*x^9 + 19*x^8 - 25*x^7 + 62*x^6 - 13*x^5 + 55*x^4 - 34*x^3 + 21*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 19 x^{8} - 25 x^{7} + 62 x^{6} - 13 x^{5} + 55 x^{4} - 34 x^{3} + 21 x^{2} - 5 x + 1 \)

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:  \(342648531949824=2^{8}\cdot 3^{14}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.26$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{30} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{30} a^{6} - \frac{1}{3} a^{5} + \frac{13}{30} a^{3} + \frac{1}{3} a - \frac{11}{30}$, $\frac{1}{30} a^{10} + \frac{1}{3} a^{8} - \frac{3}{10} a^{7} - \frac{1}{3} a^{5} + \frac{13}{30} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{30} a + \frac{1}{3}$, $\frac{1}{8460} a^{11} + \frac{17}{4230} a^{10} + \frac{67}{8460} a^{9} - \frac{2459}{8460} a^{8} + \frac{229}{1410} a^{7} - \frac{3823}{8460} a^{6} - \frac{353}{940} a^{5} - \frac{307}{2115} a^{4} + \frac{1631}{8460} a^{3} + \frac{41}{940} a^{2} + \frac{46}{235} a + \frac{2683}{8460}$

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:  \( -\frac{207}{940} a^{11} + \frac{10}{47} a^{10} - \frac{3161}{2820} a^{9} + \frac{6119}{2820} a^{8} - \frac{593}{141} a^{7} + \frac{15529}{2820} a^{6} - \frac{38683}{2820} a^{5} + \frac{237}{94} a^{4} - \frac{36473}{2820} a^{3} + \frac{5397}{940} a^{2} - \frac{1421}{282} a + \frac{3391}{2820} \) (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:  \( 1174.48503862 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T183:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for S_6(12)
Character table for S_6(12)

Intermediate fields

\(\Q(\sqrt{-3}) \), 6.4.2056752.1

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

Sibling fields

Degree 6 siblings: 6.4.2056752.1, 6.0.198291974508.1
Degree 10 sibling: Deg 10
Degree 12 sibling: Deg 12
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$23$23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.6.4.1$x^{6} + 460 x^{3} + 181447$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$