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Properties

Label 12.0.31701123988...1984.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 3^{6}\cdot 7^{10}\cdot 13^{11}$
Root discriminant $619.09$
Ramified primes $2, 3, 7, 13$
Class number $11354304$ (GRH)
Class group $[2, 2, 2, 2, 709644]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17178086016, 0, 13314502656, 0, 889945056, 0, 22407840, 0, 247338, 0, 1092, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 1092*x^10 + 247338*x^8 + 22407840*x^6 + 889945056*x^4 + 13314502656*x^2 + 17178086016)
 
gp: K = bnfinit(x^12 + 1092*x^10 + 247338*x^8 + 22407840*x^6 + 889945056*x^4 + 13314502656*x^2 + 17178086016, 1)
 

Normalized defining polynomial

\( x^{12} + 1092 x^{10} + 247338 x^{8} + 22407840 x^{6} + 889945056 x^{4} + 13314502656 x^{2} + 17178086016 \)

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:  \(3170112398857312997665091547561984=2^{33}\cdot 3^{6}\cdot 7^{10}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $619.09$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4368=2^{4}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4368}(1,·)$, $\chi_{4368}(2309,·)$, $\chi_{4368}(289,·)$, $\chi_{4368}(2957,·)$, $\chi_{4368}(2813,·)$, $\chi_{4368}(529,·)$, $\chi_{4368}(3365,·)$, $\chi_{4368}(3481,·)$, $\chi_{4368}(2521,·)$, $\chi_{4368}(509,·)$, $\chi_{4368}(1369,·)$, $\chi_{4368}(2789,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18} a^{4}$, $\frac{1}{18} a^{5}$, $\frac{1}{756} a^{6} - \frac{1}{6} a^{2}$, $\frac{1}{1512} a^{7} - \frac{1}{12} a^{3}$, $\frac{1}{72576} a^{8} - \frac{1}{2016} a^{6} + \frac{5}{192} a^{4} + \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{2467584} a^{9} - \frac{1}{68544} a^{7} - \frac{1}{1152} a^{5} + \frac{83}{816} a^{3} - \frac{73}{272} a$, $\frac{1}{259530313738752} a^{10} - \frac{15629905}{21627526144896} a^{8} - \frac{395265719}{848138280192} a^{6} - \frac{223517939}{42911758224} a^{4} + \frac{1249389317}{9535946272} a^{2} + \frac{12412427}{70117252}$, $\frac{1}{259530313738752} a^{11} + \frac{5653}{64367637336} a^{9} + \frac{798575179}{4806116921088} a^{7} - \frac{1192070077}{171647032896} a^{5} - \frac{784010991}{9535946272} a^{3} - \frac{857617331}{2383986568} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{709644}$, which has order $11354304$ (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:  \( 280406.634104081 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{26}) \), 3.3.8281.1, 4.0.1984260096.4, 6.6.456434940416.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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$2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
$3$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$7$7.12.10.4$x^{12} - 7 x^{6} + 147$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$13$13.12.11.11$x^{12} + 6656$$12$$1$$11$$C_{12}$$[\ ]_{12}$