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Properties

Label 12.0.1283918464548864.27
Degree $12$
Signature $[0, 6]$
Discriminant $2^{28}\cdot 3^{14}$
Root discriminant $18.16$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3\wr C_2$ (as 12T77)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -48, 108, -76, 9, 108, -46, -12, 45, -8, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 6*x^10 - 8*x^9 + 45*x^8 - 12*x^7 - 46*x^6 + 108*x^5 + 9*x^4 - 76*x^3 + 108*x^2 - 48*x + 16)
 
gp: K = bnfinit(x^12 + 6*x^10 - 8*x^9 + 45*x^8 - 12*x^7 - 46*x^6 + 108*x^5 + 9*x^4 - 76*x^3 + 108*x^2 - 48*x + 16, 1)
 

Normalized defining polynomial

\( x^{12} + 6 x^{10} - 8 x^{9} + 45 x^{8} - 12 x^{7} - 46 x^{6} + 108 x^{5} + 9 x^{4} - 76 x^{3} + 108 x^{2} - 48 x + 16 \)

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:  \(1283918464548864=2^{28}\cdot 3^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{144} a^{10} - \frac{1}{72} a^{9} - \frac{5}{72} a^{8} + \frac{1}{36} a^{7} - \frac{1}{48} a^{6} + \frac{5}{72} a^{5} - \frac{1}{24} a^{4} + \frac{1}{9} a^{3} + \frac{1}{144} a^{2} - \frac{7}{72} a + \frac{17}{36}$, $\frac{1}{6161760} a^{11} - \frac{5123}{3080880} a^{10} - \frac{10961}{342320} a^{9} - \frac{12335}{102696} a^{8} + \frac{32633}{1232352} a^{7} + \frac{20209}{280080} a^{6} - \frac{263407}{3080880} a^{5} - \frac{273811}{770220} a^{4} + \frac{1069697}{6161760} a^{3} - \frac{541529}{3080880} a^{2} + \frac{36351}{171160} a + \frac{99251}{385110}$

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:  \( -\frac{2491}{85580} a^{11} - \frac{1449}{85580} a^{10} - \frac{8311}{42790} a^{9} + \frac{3143}{25674} a^{8} - \frac{22035}{17116} a^{7} - \frac{2203}{7780} a^{6} + \frac{100511}{128370} a^{5} - \frac{101781}{42790} a^{4} - \frac{81927}{85580} a^{3} + \frac{327649}{256740} a^{2} - \frac{60309}{21395} a + \frac{25516}{21395} \) (order $6$)
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:  \( 1590.92800548 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\wr C_2$ (as 12T77):

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 $C_2\times S_3\wr C_2$
Character table for $C_2\times S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 6.4.11943936.1

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 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.22.9$x^{8} + 2 x^{4} + 4$$4$$2$$22$$D_4\times C_2$$[2, 3, 4]^{2}$
$3$3.12.14.5$x^{12} - 12 x^{11} - 3 x^{10} - 9 x^{7} + 9 x^{5} + 9 x^{2} + 9$$6$$2$$14$$S_3^2$$[3/2, 3/2]_{2}^{2}$