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Properties

Label 12.0.18296540302...0000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{9}\cdot 73^{11}$
Root discriminant $591.38$
Ramified primes $2, 3, 5, 73$
Class number $21962048$ (GRH)
Class group $[2, 2, 2, 2, 1372628]$ (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![15971752125, 0, 3368192625, 0, 262241550, 0, 9460800, 0, 160965, 0, 1095, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 1095*x^10 + 160965*x^8 + 9460800*x^6 + 262241550*x^4 + 3368192625*x^2 + 15971752125)
 
gp: K = bnfinit(x^12 + 1095*x^10 + 160965*x^8 + 9460800*x^6 + 262241550*x^4 + 3368192625*x^2 + 15971752125, 1)
 

Normalized defining polynomial

\( x^{12} + 1095 x^{10} + 160965 x^{8} + 9460800 x^{6} + 262241550 x^{4} + 3368192625 x^{2} + 15971752125 \)

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:  \(1829654030234673819254664000000000=2^{12}\cdot 3^{6}\cdot 5^{9}\cdot 73^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $591.38$
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:  $2, 3, 5, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4380=2^{2}\cdot 3\cdot 5\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{4380}(1,·)$, $\chi_{4380}(3649,·)$, $\chi_{4380}(649,·)$, $\chi_{4380}(1703,·)$, $\chi_{4380}(1487,·)$, $\chi_{4380}(721,·)$, $\chi_{4380}(3623,·)$, $\chi_{4380}(1463,·)$, $\chi_{4380}(3001,·)$, $\chi_{4380}(3407,·)$, $\chi_{4380}(2929,·)$, $\chi_{4380}(3647,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{45} a^{4}$, $\frac{1}{45} a^{5}$, $\frac{1}{135} a^{6}$, $\frac{1}{945} a^{7} + \frac{3}{7} a$, $\frac{1}{297675} a^{8} - \frac{2}{945} a^{6} + \frac{2}{315} a^{4} + \frac{8}{49} a^{2} + \frac{1}{3}$, $\frac{1}{297675} a^{9} + \frac{2}{315} a^{5} + \frac{8}{49} a^{3} + \frac{4}{21} a$, $\frac{1}{553165582725} a^{10} - \frac{48301}{61462842525} a^{8} + \frac{1749539}{585360405} a^{6} + \frac{6506102}{1365840945} a^{4} - \frac{26594168}{273168189} a^{2} + \frac{152441}{619429}$, $\frac{1}{3872159079075} a^{11} + \frac{190376}{143413299225} a^{9} - \frac{56201}{117072081} a^{7} - \frac{88885964}{9560886615} a^{5} + \frac{107202496}{1912177323} a^{3} - \frac{466988}{4336003} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1372628}$, which has order $21962048$ (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:  \( 196804.70719979244 \) (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

\(\Q(\sqrt{365}) \), 3.3.5329.1, 4.0.7002306000.2, 6.6.259133949125.1

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 ${\href{/LocalNumberField/7.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.12.9.3$x^{12} - 25 x^{4} + 250$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$73$73.12.11.8$x^{12} + 9125$$12$$1$$11$$C_{12}$$[\ ]_{12}$