↑

Properties

Label 12.2.346634844681273344.2
Degree $12$
Signature $[2, 5]$
Discriminant $-\,2^{33}\cdot 7^{9}$
Root discriminant $28.95$
Ramified primes $2, 7$
Class number $3$
Class group $[3]$
Galois group $C_3:S_3.D_4$ (as 12T82)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-224, 0, -896, 0, 448, 0, -48, 0, 10, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^10 + 10*x^8 - 48*x^6 + 448*x^4 - 896*x^2 - 224)
 
gp: K = bnfinit(x^12 - 4*x^10 + 10*x^8 - 48*x^6 + 448*x^4 - 896*x^2 - 224, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{10} + 10 x^{8} - 48 x^{6} + 448 x^{4} - 896 x^{2} - 224 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-346634844681273344=-\,2^{33}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.95$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{13}{64} a^{4} - \frac{5}{16} a^{2} - \frac{1}{2} a - \frac{7}{16}$, $\frac{1}{256} a^{9} - \frac{1}{32} a^{7} - \frac{1}{16} a^{6} - \frac{13}{128} a^{5} - \frac{1}{8} a^{4} + \frac{11}{32} a^{3} + \frac{3}{8} a^{2} + \frac{9}{32} a + \frac{1}{4}$, $\frac{1}{13312} a^{10} + \frac{17}{6656} a^{8} - \frac{1}{32} a^{7} - \frac{181}{6656} a^{6} + \frac{1}{16} a^{5} - \frac{123}{3328} a^{4} + \frac{3}{16} a^{3} - \frac{201}{1664} a^{2} + \frac{3}{8} a + \frac{285}{832}$, $\frac{1}{13312} a^{11} - \frac{9}{6656} a^{9} + \frac{27}{6656} a^{7} + \frac{215}{3328} a^{5} - \frac{1}{4} a^{4} - \frac{773}{1664} a^{3} + \frac{51}{832} a - \frac{1}{2}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
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:  \( 8520.23148869 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3.D_4$ (as 12T82):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_3:S_3.D_4$
Character table for $C_3:S_3.D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.14336.2, 6.2.39337984.1

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 24 siblings: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$7$7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$