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Properties

Label 12.0.35549166398...0000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}\cdot 5^{9}\cdot 13^{11}$
Root discriminant $515.90$
Ramified primes $2, 3, 5, 13$
Class number $11739360$ (GRH)
Class group $[2, 2, 2, 1467420]$ (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![3833856000, 0, 4672512000, 0, 1451174400, 0, 29335800, 0, 226980, 0, 780, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 780*x^10 + 226980*x^8 + 29335800*x^6 + 1451174400*x^4 + 4672512000*x^2 + 3833856000)
 
gp: K = bnfinit(x^12 + 780*x^10 + 226980*x^8 + 29335800*x^6 + 1451174400*x^4 + 4672512000*x^2 + 3833856000, 1)
 

Normalized defining polynomial

\( x^{12} + 780 x^{10} + 226980 x^{8} + 29335800 x^{6} + 1451174400 x^{4} + 4672512000 x^{2} + 3833856000 \)

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:  \(355491663986814578735616000000000=2^{18}\cdot 3^{18}\cdot 5^{9}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $515.90$
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:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1,·)$, $\chi_{4680}(2401,·)$, $\chi_{4680}(3721,·)$, $\chi_{4680}(649,·)$, $\chi_{4680}(1997,·)$, $\chi_{4680}(4253,·)$, $\chi_{4680}(49,·)$, $\chi_{4680}(2333,·)$, $\chi_{4680}(4373,·)$, $\chi_{4680}(4489,·)$, $\chi_{4680}(2477,·)$, $\chi_{4680}(3677,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{20} a^{4}$, $\frac{1}{20} a^{5}$, $\frac{1}{840} a^{6} - \frac{1}{7}$, $\frac{1}{3360} a^{7} - \frac{1}{40} a^{5} + \frac{1}{8} a^{3} + \frac{13}{28} a$, $\frac{1}{134400} a^{8} - \frac{1}{6720} a^{6} + \frac{1}{320} a^{4} - \frac{3}{224} a^{2} + \frac{1}{7}$, $\frac{1}{537600} a^{9} - \frac{1}{26880} a^{7} - \frac{31}{1280} a^{5} - \frac{115}{896} a^{3} + \frac{1}{28} a$, $\frac{1}{861291110400} a^{10} - \frac{732269}{215322777600} a^{8} + \frac{509191}{14354851840} a^{6} - \frac{142088367}{7177425920} a^{4} - \frac{25752}{1401841} a^{2} - \frac{427248}{1401841}$, $\frac{1}{3445164441600} a^{11} - \frac{732269}{861291110400} a^{9} + \frac{509191}{57419407360} a^{7} - \frac{500959663}{28709703680} a^{5} - \frac{1453345}{11214728} a^{3} + \frac{974593}{5607364} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{1467420}$, which has order $11739360$ (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:  \( 60875.97226910356 \) (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

\(\Q(\sqrt{65}) \), 3.3.13689.2, 4.0.158184000.1, 6.6.304506671625.2

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 R ${\href{/LocalNumberField/7.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.12.18.88$x^{12} + 21 x^{11} + 21 x^{10} - 39 x^{9} + 9 x^{8} - 36 x^{7} - 3 x^{6} + 18 x^{5} + 27 x^{4} - 27 x - 36$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$5$5.12.9.4$x^{12} + 30 x^{8} + 275 x^{4} + 1000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$13$13.12.11.9$x^{12} + 416$$12$$1$$11$$C_{12}$$[\ ]_{12}$