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Properties

Label 12.0.12486332411...4224.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 23^{6}$
Root discriminant $84.08$
Ramified primes $2, 3, 7, 23$
Class number $17472$ (GRH)
Class group $[4, 4368]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![148035889, 0, 102981488, 0, 16790460, 0, 949026, 0, 23276, 0, 253, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 253*x^10 + 23276*x^8 + 949026*x^6 + 16790460*x^4 + 102981488*x^2 + 148035889)
 
gp: K = bnfinit(x^12 + 253*x^10 + 23276*x^8 + 949026*x^6 + 16790460*x^4 + 102981488*x^2 + 148035889, 1)
 

Normalized defining polynomial

\( x^{12} + 253 x^{10} + 23276 x^{8} + 949026 x^{6} + 16790460 x^{4} + 102981488 x^{2} + 148035889 \)

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:  \(124863324110553424564224=2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.08$
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:  $2, 3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1932=2^{2}\cdot 3\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1932}(367,·)$, $\chi_{1932}(1,·)$, $\chi_{1932}(643,·)$, $\chi_{1932}(1381,·)$, $\chi_{1932}(1195,·)$, $\chi_{1932}(461,·)$, $\chi_{1932}(1103,·)$, $\chi_{1932}(275,·)$, $\chi_{1932}(277,·)$, $\chi_{1932}(185,·)$, $\chi_{1932}(827,·)$, $\chi_{1932}(1013,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{23} a^{2}$, $\frac{1}{23} a^{3}$, $\frac{1}{529} a^{4}$, $\frac{1}{529} a^{5}$, $\frac{1}{12167} a^{6}$, $\frac{1}{12167} a^{7}$, $\frac{1}{279841} a^{8}$, $\frac{1}{279841} a^{9}$, $\frac{1}{6436343} a^{10}$, $\frac{1}{6436343} a^{11}$

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

Class group and class number

$C_{4}\times C_{4368}$, which has order $17472$ (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:  \( \frac{1}{12167} a^{6} + \frac{6}{529} a^{4} + \frac{9}{23} a^{2} + 2 \),  \( \frac{1}{6436343} a^{10} + \frac{10}{279841} a^{8} + \frac{35}{12167} a^{6} + \frac{51}{529} a^{4} + \frac{29}{23} a^{2} + 4 \),  \( \frac{1}{529} a^{4} + \frac{4}{23} a^{2} + 1 \),  \( \frac{1}{23} a^{2} + 1 \),  \( \frac{1}{12167} a^{6} + \frac{6}{529} a^{4} + \frac{8}{23} a^{2} + 1 \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140.7987960054707 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-161}) \), \(\Q(\sqrt{-69}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-69})\), \(\Q(\zeta_{21})^+\), 6.0.13087409216.1, 6.0.50480006976.3

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$23$23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$