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Properties

Label 12.0.56466838261...0000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 5^{6}\cdot 13^{10}$
Root discriminant $53.62$
Ramified primes $2, 5, 13$
Class number $1344$ (GRH)
Class group $[4, 4, 84]$ (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![2641211, -377314, 1022917, -126884, 182902, -19260, 19320, -1658, 1279, -82, 51, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 51*x^10 - 82*x^9 + 1279*x^8 - 1658*x^7 + 19320*x^6 - 19260*x^5 + 182902*x^4 - 126884*x^3 + 1022917*x^2 - 377314*x + 2641211)
 
gp: K = bnfinit(x^12 - 2*x^11 + 51*x^10 - 82*x^9 + 1279*x^8 - 1658*x^7 + 19320*x^6 - 19260*x^5 + 182902*x^4 - 126884*x^3 + 1022917*x^2 - 377314*x + 2641211, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 51 x^{10} - 82 x^{9} + 1279 x^{8} - 1658 x^{7} + 19320 x^{6} - 19260 x^{5} + 182902 x^{4} - 126884 x^{3} + 1022917 x^{2} - 377314 x + 2641211 \)

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:  \(564668382613504000000=2^{18}\cdot 5^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.62$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(520=2^{3}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(259,·)$, $\chi_{520}(321,·)$, $\chi_{520}(361,·)$, $\chi_{520}(339,·)$, $\chi_{520}(139,·)$, $\chi_{520}(419,·)$, $\chi_{520}(81,·)$, $\chi_{520}(179,·)$, $\chi_{520}(121,·)$, $\chi_{520}(441,·)$, $\chi_{520}(459,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{4726025247677171193938573} a^{11} - \frac{1646582851772919332823893}{4726025247677171193938573} a^{10} - \frac{1099607399058380778315301}{4726025247677171193938573} a^{9} + \frac{405373816350785742619804}{4726025247677171193938573} a^{8} + \frac{2235781347682556172627200}{4726025247677171193938573} a^{7} - \frac{609752446228183417850842}{4726025247677171193938573} a^{6} - \frac{1482702199335256587374205}{4726025247677171193938573} a^{5} - \frac{2197076442500215337294816}{4726025247677171193938573} a^{4} - \frac{424607971037835863907810}{4726025247677171193938573} a^{3} - \frac{1747930381299227414437101}{4726025247677171193938573} a^{2} - \frac{1500826911994247479347955}{4726025247677171193938573} a - \frac{2086791163490717169381465}{4726025247677171193938573}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{84}$, which has order $1344$ (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:  \( 120.78403136265631 \) (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

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-130}) \), 3.3.169.1, \(\Q(\sqrt{-10}, \sqrt{13})\), 6.0.1827904000.2, \(\Q(\zeta_{13})^+\), 6.0.23762752000.2

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$$[3]^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$