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Properties

Label 12.0.469804094334435328.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 13^{11}$
Root discriminant $29.69$
Ramified primes $2, 13$
Class number $4$
Class group $[4]$
Galois group $C_2^5.C_6$ (as 12T99)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 56, -122, 965, -982, 1052, -600, 334, -112, 36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 36*x^10 - 112*x^9 + 334*x^8 - 600*x^7 + 1052*x^6 - 982*x^5 + 965*x^4 - 122*x^3 + 56*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^12 - 6*x^11 + 36*x^10 - 112*x^9 + 334*x^8 - 600*x^7 + 1052*x^6 - 982*x^5 + 965*x^4 - 122*x^3 + 56*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 36 x^{10} - 112 x^{9} + 334 x^{8} - 600 x^{7} + 1052 x^{6} - 982 x^{5} + 965 x^{4} - 122 x^{3} + 56 x^{2} + 2 x + 1 \)

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:  \(469804094334435328=2^{18}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.69$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{51498595} a^{11} - \frac{5021811}{10299719} a^{10} - \frac{1641254}{51498595} a^{9} + \frac{18419244}{51498595} a^{8} + \frac{24079038}{51498595} a^{7} + \frac{59738}{51498595} a^{6} - \frac{3258028}{10299719} a^{5} + \frac{15251603}{51498595} a^{4} - \frac{12898722}{51498595} a^{3} - \frac{21219744}{51498595} a^{2} - \frac{13420783}{51498595} a - \frac{18360931}{51498595}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3216.07312502 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_6$ (as 12T99):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 20 conjugacy class representatives for $C_2^5.C_6$
Character table for $C_2^5.C_6$

Intermediate fields

3.3.169.1, 6.2.23762752.1

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 16 siblings: 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.18.36$x^{12} - 6 x^{11} + 6 x^{10} + 8 x^{9} + 8 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8$$4$$3$$18$12T99$[2, 2, 2, 2, 2]^{6}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$