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Properties

Label 12.4.1263657015625.1
Degree $12$
Signature $[4, 4]$
Discriminant $5^{6}\cdot 17^{2}\cdot 23^{4}$
Root discriminant $10.20$
Ramified primes $5, 17, 23$
Class number $1$
Class group Trivial
Galois group $C_2^2\times S_4$ (as 12T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 13, -7, 20, -11, 11, -1, 2, 1, -3, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - x^10 - 3*x^9 + x^8 + 2*x^7 - x^6 + 11*x^5 - 11*x^4 + 20*x^3 - 7*x^2 + 13*x + 1)
 
gp: K = bnfinit(x^12 - x^11 - x^10 - 3*x^9 + x^8 + 2*x^7 - x^6 + 11*x^5 - 11*x^4 + 20*x^3 - 7*x^2 + 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - x^{10} - 3 x^{9} + x^{8} + 2 x^{7} - x^{6} + 11 x^{5} - 11 x^{4} + 20 x^{3} - 7 x^{2} + 13 x + 1 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1263657015625=5^{6}\cdot 17^{2}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.20$
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:  $5, 17, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{1075} a^{11} - \frac{8}{215} a^{10} + \frac{54}{1075} a^{9} + \frac{41}{1075} a^{8} - \frac{93}{1075} a^{7} - \frac{26}{1075} a^{6} - \frac{62}{1075} a^{5} - \frac{151}{1075} a^{4} + \frac{503}{1075} a^{3} - \frac{32}{1075} a^{2} + \frac{381}{1075} a - \frac{226}{1075}$

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:  $7$
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:  \( 15.3002273809 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times S_4$ (as 12T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 20 conjugacy class representatives for $C_2^2\times S_4$
Character table for $C_2^2\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.23.1, 6.2.224825.1, 6.2.44965.1, 6.2.66125.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$