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Properties

Label 12.0.20833285483...3056.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 7^{8}\cdot 13^{10}$
Root discriminant $87.75$
Ramified primes $2, 7, 13$
Class number $12663$ (GRH)
Class group $[3, 4221]$ (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![250587, -100962, 140349, -34388, 18694, 9584, -5152, -318, 975, 50, -49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 49*x^10 + 50*x^9 + 975*x^8 - 318*x^7 - 5152*x^6 + 9584*x^5 + 18694*x^4 - 34388*x^3 + 140349*x^2 - 100962*x + 250587)
 
gp: K = bnfinit(x^12 - 2*x^11 - 49*x^10 + 50*x^9 + 975*x^8 - 318*x^7 - 5152*x^6 + 9584*x^5 + 18694*x^4 - 34388*x^3 + 140349*x^2 - 100962*x + 250587, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 49 x^{10} + 50 x^{9} + 975 x^{8} - 318 x^{7} - 5152 x^{6} + 9584 x^{5} + 18694 x^{4} - 34388 x^{3} + 140349 x^{2} - 100962 x + 250587 \)

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:  \(208332854832557470253056=2^{18}\cdot 7^{8}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.75$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(627,·)$, $\chi_{728}(1,·)$, $\chi_{728}(547,·)$, $\chi_{728}(667,·)$, $\chi_{728}(337,·)$, $\chi_{728}(9,·)$, $\chi_{728}(555,·)$, $\chi_{728}(81,·)$, $\chi_{728}(179,·)$, $\chi_{728}(361,·)$, $\chi_{728}(121,·)$, $\chi_{728}(155,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{27} a^{5} + \frac{1}{27} a^{4} - \frac{2}{27} a^{3} - \frac{11}{27} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{81} a^{8} + \frac{1}{81} a^{7} + \frac{4}{81} a^{6} + \frac{1}{81} a^{5} + \frac{1}{81} a^{4} - \frac{11}{81} a^{3} - \frac{8}{27} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{243} a^{9} - \frac{1}{243} a^{8} + \frac{2}{243} a^{7} + \frac{11}{243} a^{6} + \frac{8}{243} a^{5} - \frac{40}{243} a^{4} + \frac{16}{243} a^{3} - \frac{2}{81} a^{2} - \frac{1}{27} a + \frac{2}{9}$, $\frac{1}{2187} a^{10} + \frac{4}{2187} a^{9} + \frac{1}{243} a^{8} - \frac{1}{729} a^{7} + \frac{16}{729} a^{6} + \frac{37}{729} a^{5} + \frac{305}{2187} a^{4} + \frac{122}{2187} a^{3} - \frac{193}{729} a^{2} - \frac{47}{243} a + \frac{1}{81}$, $\frac{1}{56289439183282479} a^{11} - \frac{9998803712636}{56289439183282479} a^{10} + \frac{10023625420007}{18763146394427493} a^{9} - \frac{74548288750777}{18763146394427493} a^{8} - \frac{65917865299325}{18763146394427493} a^{7} + \frac{732160647664777}{18763146394427493} a^{6} - \frac{268756473538495}{56289439183282479} a^{5} - \frac{5108436379606951}{56289439183282479} a^{4} - \frac{441554551548802}{6254382131475831} a^{3} + \frac{2328464359821716}{6254382131475831} a^{2} - \frac{15394168662262}{694931347941759} a + \frac{205779282490586}{694931347941759}$

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

Class group and class number

$C_{3}\times C_{4221}$, which has order $12663$ (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:  \( 17569.03784509895 \) (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

\(\Q(\sqrt{-26}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-2}) \), 3.3.8281.2, \(\Q(\sqrt{-2}, \sqrt{13})\), 6.0.456434940416.5, 6.6.891474493.2, 6.0.35110380032.4

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$7$7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$