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Properties

Label 12.0.11968030099...0000.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 13^{6}$
Root discriminant $83.79$
Ramified primes $2, 3, 5, 13$
Class number $19152$ (GRH)
Class group $[6, 3192]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42211009, 0, 11367462, 0, 1286037, 0, 81283, 0, 3228, 0, 81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 81*x^10 + 3228*x^8 + 81283*x^6 + 1286037*x^4 + 11367462*x^2 + 42211009)
 
gp: K = bnfinit(x^12 + 81*x^10 + 3228*x^8 + 81283*x^6 + 1286037*x^4 + 11367462*x^2 + 42211009, 1)
 

Normalized defining polynomial

\( x^{12} + 81 x^{10} + 3228 x^{8} + 81283 x^{6} + 1286037 x^{4} + 11367462 x^{2} + 42211009 \)

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:  \(119680300997734464000000=2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.79$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2340=2^{2}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2340}(1,·)$, $\chi_{2340}(259,·)$, $\chi_{2340}(911,·)$, $\chi_{2340}(389,·)$, $\chi_{2340}(1819,·)$, $\chi_{2340}(781,·)$, $\chi_{2340}(1039,·)$, $\chi_{2340}(1169,·)$, $\chi_{2340}(131,·)$, $\chi_{2340}(1561,·)$, $\chi_{2340}(1691,·)$, $\chi_{2340}(1949,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{121567358071} a^{10} - \frac{27506895281}{121567358071} a^{8} - \frac{60192725156}{121567358071} a^{6} + \frac{60009262946}{121567358071} a^{4} + \frac{31861811008}{121567358071} a^{2} + \frac{27810413723}{121567358071}$, $\frac{1}{789823125387287} a^{11} + \frac{186821522459846}{789823125387287} a^{9} + \frac{259364549398358}{789823125387287} a^{7} + \frac{189583520495635}{789823125387287} a^{5} + \frac{89991706783548}{789823125387287} a^{3} + \frac{132900932785326}{789823125387287} a$

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

Class group and class number

$C_{6}\times C_{3192}$, which has order $19152$ (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:  $5$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 325.67540279491664 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-65}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-65})\), 6.0.115316136000.11, 6.0.5405443875.3, \(\Q(\zeta_{36})^+\)

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

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$