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Properties

Label 12.12.3167202235...0625.2
Degree $12$
Signature $[12, 0]$
Discriminant $5^{6}\cdot 7^{8}\cdot 181^{6}$
Root discriminant $110.08$
Ramified primes $5, 7, 181$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^2 \times A_4$ (as 12T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2784763981, 473494266, -547182535, -54271858, 39942353, 2399916, -1436344, -50941, 27283, 517, -262, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 262*x^10 + 517*x^9 + 27283*x^8 - 50941*x^7 - 1436344*x^6 + 2399916*x^5 + 39942353*x^4 - 54271858*x^3 - 547182535*x^2 + 473494266*x + 2784763981)
 
gp: K = bnfinit(x^12 - 2*x^11 - 262*x^10 + 517*x^9 + 27283*x^8 - 50941*x^7 - 1436344*x^6 + 2399916*x^5 + 39942353*x^4 - 54271858*x^3 - 547182535*x^2 + 473494266*x + 2784763981, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 262 x^{10} + 517 x^{9} + 27283 x^{8} - 50941 x^{7} - 1436344 x^{6} + 2399916 x^{5} + 39942353 x^{4} - 54271858 x^{3} - 547182535 x^{2} + 473494266 x + 2784763981 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3167202235965388685640625=5^{6}\cdot 7^{8}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.08$
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:  $5, 7, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{10} + \frac{2}{25} a^{8} - \frac{4}{25} a^{7} + \frac{3}{25} a^{6} + \frac{9}{25} a^{5} - \frac{9}{25} a^{4} - \frac{1}{25} a^{3} + \frac{3}{25} a - \frac{4}{25}$, $\frac{1}{14032388694620054203320447101275} a^{11} - \frac{160362222702898950714410280566}{14032388694620054203320447101275} a^{10} + \frac{251717316145000298954498888092}{14032388694620054203320447101275} a^{9} - \frac{313845926742496846929115768421}{14032388694620054203320447101275} a^{8} - \frac{3237445706889051372823444226438}{14032388694620054203320447101275} a^{7} + \frac{558524925500198271256733770096}{14032388694620054203320447101275} a^{6} + \frac{371447823381389175132048151977}{14032388694620054203320447101275} a^{5} - \frac{3516458092941269236652436972467}{14032388694620054203320447101275} a^{4} - \frac{2069871374841775688527400938479}{14032388694620054203320447101275} a^{3} - \frac{3101824577532985598606547054432}{14032388694620054203320447101275} a^{2} + \frac{448228139822660511355082928063}{14032388694620054203320447101275} a - \frac{6970681921729607551891740363426}{14032388694620054203320447101275}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $11$
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:  \( 20899401.0687 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times A_4$ (as 12T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 16 conjugacy class representatives for $C_2^2 \times A_4$
Character table for $C_2^2 \times A_4$

Intermediate fields

\(\Q(\sqrt{905}) \), \(\Q(\zeta_{7})^+\), 6.6.1779663517625.1, 6.6.9832395125.1, 6.6.434581.1

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

Sibling fields

Degree 12 siblings: 12.12.2950947586890625.1, 12.12.96675993894123765625.1, 12.12.96675993894123765625.2, 12.12.3167202235965388685640625.1
Degree 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$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}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
181Data not computed