↑

Properties

Label 12.0.29535382859...1328.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 3^{18}\cdot 31^{6}$
Root discriminant $194.62$
Ramified primes $2, 3, 31$
Class number $1295944$ (GRH)
Class group $[2, 2, 323986]$ (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63900265032, 0, 12367793232, 0, 698181876, 0, 16444632, 0, 172980, 0, 744, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 744*x^10 + 172980*x^8 + 16444632*x^6 + 698181876*x^4 + 12367793232*x^2 + 63900265032)
 
gp: K = bnfinit(x^12 + 744*x^10 + 172980*x^8 + 16444632*x^6 + 698181876*x^4 + 12367793232*x^2 + 63900265032, 1)
 

Normalized defining polynomial

\( x^{12} + 744 x^{10} + 172980 x^{8} + 16444632 x^{6} + 698181876 x^{4} + 12367793232 x^{2} + 63900265032 \)

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:  \(2953538285909432612922851328=2^{33}\cdot 3^{18}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $194.62$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4464=2^{4}\cdot 3^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{4464}(1,·)$, $\chi_{4464}(1859,·)$, $\chi_{4464}(2977,·)$, $\chi_{4464}(3721,·)$, $\chi_{4464}(2603,·)$, $\chi_{4464}(1489,·)$, $\chi_{4464}(1115,·)$, $\chi_{4464}(3347,·)$, $\chi_{4464}(745,·)$, $\chi_{4464}(371,·)$, $\chi_{4464}(2233,·)$, $\chi_{4464}(4091,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{31} a^{2}$, $\frac{1}{31} a^{3}$, $\frac{1}{1922} a^{4}$, $\frac{1}{1922} a^{5}$, $\frac{1}{178746} a^{6}$, $\frac{1}{178746} a^{7}$, $\frac{1}{11082252} a^{8}$, $\frac{1}{11082252} a^{9}$, $\frac{1}{5840346804} a^{10} - \frac{7}{188398284} a^{8} + \frac{1}{1012894} a^{6} - \frac{7}{32674} a^{4} - \frac{7}{527} a^{2} - \frac{2}{17}$, $\frac{1}{5840346804} a^{11} - \frac{7}{188398284} a^{9} + \frac{1}{1012894} a^{7} - \frac{7}{32674} a^{5} - \frac{7}{527} a^{3} - \frac{2}{17} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{323986}$, which has order $1295944$ (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:  \( 481.70037561485367 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 4.0.17713152.5, 6.6.3359232.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.33.376$x^{12} + 36 x^{10} + 42 x^{8} - 40 x^{6} + 40 x^{4} + 32 x^{2} - 56$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$3$3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$31$31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$