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Properties

Label 12.0.5777633090469888.27
Degree $12$
Signature $[0, 6]$
Discriminant $2^{27}\cdot 3^{16}$
Root discriminant $20.58$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_3\times C_3:D_4$ (as 12T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![200, 0, -72, 0, 180, 0, -188, 0, 72, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 + 72*x^8 - 188*x^6 + 180*x^4 - 72*x^2 + 200)
 
gp: K = bnfinit(x^12 - 12*x^10 + 72*x^8 - 188*x^6 + 180*x^4 - 72*x^2 + 200, 1)
 

Normalized defining polynomial

\( x^{12} - 12 x^{10} + 72 x^{8} - 188 x^{6} + 180 x^{4} - 72 x^{2} + 200 \)

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:  \(5777633090469888=2^{27}\cdot 3^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.58$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{20} a^{9} + \frac{1}{10} a^{7} - \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{89620} a^{10} + \frac{2678}{22405} a^{8} - \frac{864}{4481} a^{6} - \frac{10779}{44810} a^{4} + \frac{7947}{22405} a^{2} - \frac{1002}{4481}$, $\frac{1}{89620} a^{11} + \frac{175}{8962} a^{9} + \frac{4803}{44810} a^{7} - \frac{10779}{44810} a^{5} + \frac{3466}{22405} a^{3} - \frac{529}{22405} a$

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:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{42}{4481} a^{10} + \frac{873}{8962} a^{8} - \frac{4805}{8962} a^{6} + \frac{4755}{4481} a^{4} - \frac{4239}{4481} a^{2} + \frac{3733}{4481} \) (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:  \( 3903.06692197 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:D_4$ (as 12T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 27 conjugacy class representatives for $C_3\times C_3:D_4$
Character table for $C_3\times C_3:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.512.1, 6.0.419904.3

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.27.303$x^{12} + 4 x^{10} - 8 x^{8} + 12 x^{6} + 4 x^{4} - 8 x^{2} + 8$$4$$3$$27$$D_4 \times C_3$$[2, 3, 7/2]^{3}$
$3$3.12.16.33$x^{12} + 105 x^{11} - 513 x^{10} - 834 x^{9} - 117 x^{8} - 459 x^{7} - 1008 x^{6} - 81 x^{5} - 270 x^{3} + 648 x^{2} - 486 x + 810$$3$$4$$16$$C_3\times (C_3 : C_4)$$[2, 2]^{4}$