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Properties

Label 12.4.5231047633534976.8
Degree $12$
Signature $[4, 4]$
Discriminant $2^{28}\cdot 11^{7}$
Root discriminant $20.41$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
Galois group 12T221

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44, 0, -32, 0, 4, 0, -20, 0, 6, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 + 6*x^8 - 20*x^6 + 4*x^4 - 32*x^2 + 44)
 
gp: K = bnfinit(x^12 - 2*x^10 + 6*x^8 - 20*x^6 + 4*x^4 - 32*x^2 + 44, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{10} + 6 x^{8} - 20 x^{6} + 4 x^{4} - 32 x^{2} + 44 \)

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:  \(5231047633534976=2^{28}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.41$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2266} a^{10} - \frac{256}{1133} a^{8} + \frac{268}{1133} a^{6} + \frac{403}{1133} a^{4} - \frac{455}{1133} a^{2} - \frac{21}{103}$, $\frac{1}{4532} a^{11} - \frac{128}{1133} a^{9} + \frac{134}{1133} a^{7} + \frac{403}{2266} a^{5} + \frac{339}{1133} a^{3} + \frac{41}{103} a$

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:  \( 2576.4477257 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T221:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1536
The 40 conjugacy class representatives for [1/2.cD(4)^3]S(3)
Character table for [1/2.cD(4)^3]S(3) is not computed

Intermediate fields

3.1.44.1, 6.2.340736.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.28.9$x^{12} + 4 x^{11} + 2 x^{10} - 4 x^{9} - 2 x^{8} + 8 x^{7} + 2 x^{6} - 4 x^{5} + 6 x^{4} + 8 x^{3} + 4 x^{2} + 8 x - 6$$12$$1$$28$12T221$[4/3, 4/3, 2, 8/3, 8/3, 3, 10/3, 10/3]_{3}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$