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Properties

Label 12.4.469804094334435328.4
Degree $12$
Signature $[4, 4]$
Discriminant $2^{18}\cdot 13^{11}$
Root discriminant $29.69$
Ramified primes $2, 13$
Class number $2$
Class group $[2]$
Galois group $C_2^5.C_6$ (as 12T99)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 0, -910, 0, -507, 0, -234, 0, -26, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 26*x^8 - 234*x^6 - 507*x^4 - 910*x^2 + 13)
 
gp: K = bnfinit(x^12 - 26*x^8 - 234*x^6 - 507*x^4 - 910*x^2 + 13, 1)
 

Normalized defining polynomial

\( x^{12} - 26 x^{8} - 234 x^{6} - 507 x^{4} - 910 x^{2} + 13 \)

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:  \(469804094334435328=2^{18}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.69$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{8} + \frac{1}{5} a^{6} + \frac{1}{10} a^{4} - \frac{1}{10} a^{2} + \frac{2}{5}$, $\frac{1}{20} a^{9} - \frac{3}{20} a^{7} - \frac{1}{4} a^{6} + \frac{1}{20} a^{5} - \frac{3}{10} a^{3} - \frac{1}{4} a^{2} + \frac{9}{20} a - \frac{1}{4}$, $\frac{1}{1083700} a^{10} + \frac{15609}{1083700} a^{8} - \frac{1}{4} a^{7} + \frac{5019}{216740} a^{6} + \frac{56313}{541850} a^{4} - \frac{1}{4} a^{3} + \frac{434067}{1083700} a^{2} - \frac{1}{4} a + \frac{56339}{541850}$, $\frac{1}{1083700} a^{11} + \frac{15609}{1083700} a^{9} - \frac{1}{20} a^{8} + \frac{5019}{216740} a^{7} - \frac{1}{10} a^{6} + \frac{56313}{541850} a^{5} - \frac{1}{20} a^{4} + \frac{434067}{1083700} a^{3} + \frac{1}{20} a^{2} + \frac{56339}{541850} a + \frac{3}{10}$

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:  $7$
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:  \( 15061.8371795 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_6$ (as 12T99):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 20 conjugacy class representatives for $C_2^5.C_6$
Character table for $C_2^5.C_6$

Intermediate fields

3.3.169.1, 6.2.23762752.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.18.58$x^{12} + 4 x^{11} + 6 x^{10} + 4 x^{9} - 4 x^{7} - 4 x^{6} + 8 x^{5} - 4 x^{4} + 8 x^{3} + 8 x^{2} + 8$$4$$3$$18$12T99$[2, 2, 2, 2, 2]^{6}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$