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Properties

Label 12.0.548345023565824.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 43^{2}\cdot 67^{2}\cdot 127^{2}$
Root discriminant $16.91$
Ramified primes $2, 43, 67, 127$
Class number $1$
Class group Trivial
Galois group $S_6\times C_2$ (as 12T219)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -128, 112, -64, 40, -44, 41, -22, 10, -8, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 7*x^10 - 8*x^9 + 10*x^8 - 22*x^7 + 41*x^6 - 44*x^5 + 40*x^4 - 64*x^3 + 112*x^2 - 128*x + 64)
 
gp: K = bnfinit(x^12 - 4*x^11 + 7*x^10 - 8*x^9 + 10*x^8 - 22*x^7 + 41*x^6 - 44*x^5 + 40*x^4 - 64*x^3 + 112*x^2 - 128*x + 64, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 7 x^{10} - 8 x^{9} + 10 x^{8} - 22 x^{7} + 41 x^{6} - 44 x^{5} + 40 x^{4} - 64 x^{3} + 112 x^{2} - 128 x + 64 \)

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:  \(548345023565824=2^{12}\cdot 43^{2}\cdot 67^{2}\cdot 127^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.91$
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:  $2, 43, 67, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{48} a^{10} + \frac{1}{24} a^{9} + \frac{1}{16} a^{8} + \frac{1}{24} a^{7} + \frac{11}{24} a^{6} + \frac{1}{8} a^{5} + \frac{13}{48} a^{4} - \frac{7}{24} a^{3} - \frac{1}{4} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{96} a^{11} - \frac{1}{96} a^{9} - \frac{1}{24} a^{8} + \frac{3}{16} a^{7} + \frac{5}{48} a^{6} - \frac{47}{96} a^{5} - \frac{5}{12} a^{4} + \frac{1}{6} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a - \frac{1}{3}$

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:  \( \frac{7}{48} a^{11} - \frac{23}{48} a^{10} + \frac{31}{48} a^{9} - \frac{25}{48} a^{8} + \frac{11}{12} a^{7} - \frac{31}{12} a^{6} + \frac{181}{48} a^{5} - \frac{41}{16} a^{4} + \frac{67}{24} a^{3} - \frac{83}{12} a^{2} + 11 a - \frac{22}{3} \) (order $4$)
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:  \( 968.211843826 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_6\times C_2$ (as 12T219):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 22 conjugacy class representatives for $S_6\times C_2$
Character table for $S_6\times C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 6.6.23416768.1

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 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$127$127.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
127.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$