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Properties

Label 12.0.9845005879332009.3
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 71^{4}$
Root discriminant $21.52$
Ramified primes $3, 71$
Class number $4$
Class group $[4]$
Galois group $A_4 \times C_2$ (as 12T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3249, 2052, 4545, -228, 2799, -384, 1168, -324, 267, -32, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249)
 
gp: K = bnfinit(x^12 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249, 1)
 

Normalized defining polynomial

\( x^{12} + 18 x^{10} - 32 x^{9} + 267 x^{8} - 324 x^{7} + 1168 x^{6} - 384 x^{5} + 2799 x^{4} - 228 x^{3} + 4545 x^{2} + 2052 x + 3249 \)

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:  \(9845005879332009=3^{18}\cdot 71^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.52$
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:  $3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{8} + \frac{1}{18} a^{7} + \frac{1}{18} a^{6} - \frac{7}{18} a^{5} + \frac{5}{18} a^{4} - \frac{2}{9} a^{3} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{18} a^{9} + \frac{1}{18} a^{6} - \frac{1}{3} a^{5} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{7} + \frac{2}{9} a^{4} + \frac{1}{6} a$, $\frac{1}{7543105483554} a^{11} + \frac{54890755}{22055863987} a^{10} - \frac{29683722487}{3771552741777} a^{9} + \frac{48708538721}{2514368494518} a^{8} + \frac{46381138840}{3771552741777} a^{7} - \frac{103968134609}{2514368494518} a^{6} + \frac{1591507266322}{3771552741777} a^{5} + \frac{3439199298349}{7543105483554} a^{4} - \frac{886023392771}{3771552741777} a^{3} - \frac{5741902536}{22055863987} a^{2} + \frac{1000086484321}{2514368494518} a + \frac{54324281147}{132335183922}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{4102188952}{3771552741777} a^{11} - \frac{66531530}{66167591961} a^{10} + \frac{16329264793}{1257184247259} a^{9} - \frac{223429805149}{3771552741777} a^{8} + \frac{1680842399177}{7543105483554} a^{7} - \frac{3342446322127}{7543105483554} a^{6} + \frac{388918882651}{838122831506} a^{5} - \frac{4364133541997}{7543105483554} a^{4} + \frac{4443930623839}{7543105483554} a^{3} - \frac{86127830828}{66167591961} a^{2} - \frac{191409735199}{1257184247259} a + \frac{14420559703}{132335183922} \) (order $18$)
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:  \( 3039.60843665 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4$ (as 12T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 8 conjugacy class representatives for $A_4 \times C_2$
Character table for $A_4 \times C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 6.0.99222003.2, 6.6.33074001.1

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 8 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$