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Properties

Label 12.6.10721964631207.1
Degree $12$
Signature $[6, 3]$
Discriminant $-\,61^{2}\cdot 1423^{3}$
Root discriminant $12.19$
Ramified primes $61, 1423$
Class number $1$
Class group Trivial
Galois group 12T231

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 5, -6, 0, 7, -9, 7, -3, -7, 8, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - x^10 + 8*x^9 - 7*x^8 - 3*x^7 + 7*x^6 - 9*x^5 + 7*x^4 - 6*x^2 + 5*x - 1)
 
gp: K = bnfinit(x^12 - 2*x^11 - x^10 + 8*x^9 - 7*x^8 - 3*x^7 + 7*x^6 - 9*x^5 + 7*x^4 - 6*x^2 + 5*x - 1, 1)
 

Normalized defining polynomial

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

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10721964631207=-\,61^{2}\cdot 1423^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.19$
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:  $61, 1423$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{249} a^{11} + \frac{20}{249} a^{10} + \frac{107}{249} a^{9} + \frac{38}{249} a^{8} - \frac{1}{249} a^{7} - \frac{36}{83} a^{6} + \frac{38}{249} a^{5} - \frac{86}{249} a^{4} + \frac{107}{249} a^{3} + \frac{10}{83} a^{2} + \frac{73}{249} a - \frac{49}{249}$

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:  $8$
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:  \( 73.9804186368 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T231:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1944
The 51 conjugacy class representatives for [3^4]S(4)=3wrS(4) are not computed
Character table for [3^4]S(4)=3wrS(4) is not computed

Intermediate fields

4.2.1423.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.3.2.2$x^{3} + 122$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
1423Data not computed