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Properties

Label 12.0.14992384000000.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{16}\cdot 5^{6}\cdot 11^{4}$
Root discriminant $12.53$
Ramified primes $2, 5, 11$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_2^2$ (as 12T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 8, 8, 0, 20, 20, 0, -16, 20, -4, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 2*x^10 - 4*x^9 + 20*x^8 - 16*x^7 + 20*x^5 + 20*x^4 + 8*x^2 + 8*x + 4)
 
gp: K = bnfinit(x^12 - 2*x^11 + 2*x^10 - 4*x^9 + 20*x^8 - 16*x^7 + 20*x^5 + 20*x^4 + 8*x^2 + 8*x + 4, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 2 x^{10} - 4 x^{9} + 20 x^{8} - 16 x^{7} + 20 x^{5} + 20 x^{4} + 8 x^{2} + 8 x + 4 \)

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:  \(14992384000000=2^{16}\cdot 5^{6}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.53$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{84} a^{9} - \frac{1}{12} a^{8} + \frac{3}{14} a^{7} - \frac{1}{21} a^{6} - \frac{11}{42} a^{5} - \frac{4}{21} a^{4} - \frac{1}{42} a^{3} - \frac{11}{42} a^{2} + \frac{4}{21} a + \frac{1}{21}$, $\frac{1}{84} a^{10} - \frac{5}{42} a^{8} - \frac{1}{21} a^{7} - \frac{2}{21} a^{6} + \frac{10}{21} a^{5} - \frac{5}{14} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{8}{21} a + \frac{1}{3}$, $\frac{1}{588} a^{11} - \frac{1}{196} a^{10} - \frac{1}{294} a^{9} + \frac{1}{49} a^{8} - \frac{5}{147} a^{7} - \frac{5}{294} a^{6} + \frac{5}{294} a^{5} + \frac{5}{294} a^{4} - \frac{1}{147} a^{3} + \frac{19}{49} a^{2} + \frac{12}{49} a - \frac{34}{147}$

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:  \( -\frac{13}{49} a^{11} + \frac{57}{98} a^{10} - \frac{187}{294} a^{9} + \frac{331}{294} a^{8} - \frac{523}{98} a^{7} + \frac{740}{147} a^{6} - \frac{82}{147} a^{5} - \frac{964}{147} a^{4} - \frac{278}{147} a^{3} - \frac{37}{147} a^{2} - \frac{401}{147} a - \frac{233}{147} \) (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:  \( 66.2767777279 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times S_3$ (as 12T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 12 conjugacy class representatives for $S_3 \times C_2^2$
Character table for $S_3 \times C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), 3.1.44.1, \(\Q(i, \sqrt{5})\), 6.2.242000.1, 6.0.3872000.2, 6.0.30976.1

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 12 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$