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Properties

Label 12.0.41085390865563648.22
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 3^{14}$
Root discriminant $24.24$
Ramified primes $2, 3$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $S_3\wr C_2$ (as 12T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, -576, 888, -1152, 1090, -784, 460, -208, 91, -28, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 28*x^9 + 91*x^8 - 208*x^7 + 460*x^6 - 784*x^5 + 1090*x^4 - 1152*x^3 + 888*x^2 - 576*x + 216)
 
gp: K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 28*x^9 + 91*x^8 - 208*x^7 + 460*x^6 - 784*x^5 + 1090*x^4 - 1152*x^3 + 888*x^2 - 576*x + 216, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 10 x^{10} - 28 x^{9} + 91 x^{8} - 208 x^{7} + 460 x^{6} - 784 x^{5} + 1090 x^{4} - 1152 x^{3} + 888 x^{2} - 576 x + 216 \)

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:  \(41085390865563648=2^{33}\cdot 3^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.24$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{72} a^{10} - \frac{1}{36} a^{9} - \frac{1}{12} a^{8} + \frac{1}{36} a^{7} + \frac{5}{72} a^{6} - \frac{1}{3} a^{5} + \frac{5}{36} a^{4} + \frac{7}{18} a^{3} + \frac{1}{4} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{559091376} a^{11} + \frac{1209283}{279545688} a^{10} - \frac{3646403}{139772844} a^{9} - \frac{358373}{69886422} a^{8} - \frac{10474793}{559091376} a^{7} + \frac{51472393}{279545688} a^{6} - \frac{10376425}{279545688} a^{5} + \frac{28668305}{139772844} a^{4} - \frac{48270841}{279545688} a^{3} - \frac{16572629}{46590948} a^{2} + \frac{16649153}{46590948} a - \frac{187747}{7765158}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24461.4495528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 12T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.18432.1, 6.0.11943936.4 x3

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

Sibling fields

Degree 6 siblings: data not computed
Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
Degree 18 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.4.11.6$x^{4} + 18$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.6$x^{4} + 18$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.6$x^{4} + 18$$4$$1$$11$$D_{4}$$[2, 3, 4]$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$