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Properties

Label 12.0.17535343860...0000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 3^{6}\cdot 5^{6}\cdot 13^{11}$
Root discriminant $273.52$
Ramified primes $2, 3, 5, 13$
Class number $2032128$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 126, 252]$ (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![740390625000, 0, 60652800000, 0, 1890135000, 0, 28431000, 0, 216450, 0, 780, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 780*x^10 + 216450*x^8 + 28431000*x^6 + 1890135000*x^4 + 60652800000*x^2 + 740390625000)
 
gp: K = bnfinit(x^12 + 780*x^10 + 216450*x^8 + 28431000*x^6 + 1890135000*x^4 + 60652800000*x^2 + 740390625000, 1)
 

Normalized defining polynomial

\( x^{12} + 780 x^{10} + 216450 x^{8} + 28431000 x^{6} + 1890135000 x^{4} + 60652800000 x^{2} + 740390625000 \)

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:  \(175353438602139317305344000000=2^{33}\cdot 3^{6}\cdot 5^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $273.52$
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:  \(3120=2^{4}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3120}(1,·)$, $\chi_{3120}(2521,·)$, $\chi_{3120}(899,·)$, $\chi_{3120}(2401,·)$, $\chi_{3120}(361,·)$, $\chi_{3120}(2699,·)$, $\chi_{3120}(2579,·)$, $\chi_{3120}(2161,·)$, $\chi_{3120}(2099,·)$, $\chi_{3120}(1259,·)$, $\chi_{3120}(121,·)$, $\chi_{3120}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{15} a^{2}$, $\frac{1}{15} a^{3}$, $\frac{1}{450} a^{4}$, $\frac{1}{2250} a^{5} - \frac{2}{5} a$, $\frac{1}{33750} a^{6} - \frac{2}{75} a^{2}$, $\frac{1}{33750} a^{7} - \frac{2}{75} a^{3}$, $\frac{1}{1012500} a^{8} - \frac{1}{1125} a^{4}$, $\frac{1}{5062500} a^{9} + \frac{1}{11250} a^{5} - \frac{6}{25} a$, $\frac{1}{29995312500} a^{10} - \frac{16}{99984375} a^{8} + \frac{751}{66656250} a^{6} + \frac{134}{444375} a^{4} - \frac{2906}{148125} a^{2} + \frac{39}{79}$, $\frac{1}{29995312500} a^{11} + \frac{1}{26662500} a^{9} + \frac{751}{66656250} a^{7} - \frac{8}{148125} a^{5} - \frac{2906}{148125} a^{3} - \frac{684}{1975} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{126}\times C_{252}$, which has order $2032128$ (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:  \( 4543.270357084286 \) (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

\(\Q(\sqrt{26}) \), 3.3.169.1, 4.0.1012377600.4, 6.6.190102016.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.33.321$x^{12} + 8 x^{10} - 12 x^{8} - 24 x^{6} + 20 x^{4} - 16 x^{2} - 24$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$3$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.12.11.11$x^{12} + 6656$$12$$1$$11$$C_{12}$$[\ ]_{12}$