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Properties

Label 12.0.100498840557921.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 13^{10}$
Root discriminant $14.68$
Ramified primes $3, 13$
Class number $2$
Class group $[2]$
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![1, 3, 15, -10, 43, -5, 41, -11, 23, -3, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 6*x^10 - 3*x^9 + 23*x^8 - 11*x^7 + 41*x^6 - 5*x^5 + 43*x^4 - 10*x^3 + 15*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^12 - x^11 + 6*x^10 - 3*x^9 + 23*x^8 - 11*x^7 + 41*x^6 - 5*x^5 + 43*x^4 - 10*x^3 + 15*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 6 x^{10} - 3 x^{9} + 23 x^{8} - 11 x^{7} + 41 x^{6} - 5 x^{5} + 43 x^{4} - 10 x^{3} + 15 x^{2} + 3 x + 1 \)

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:  \(100498840557921=3^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.68$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(39=3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{39}(1,·)$, $\chi_{39}(35,·)$, $\chi_{39}(4,·)$, $\chi_{39}(38,·)$, $\chi_{39}(10,·)$, $\chi_{39}(14,·)$, $\chi_{39}(16,·)$, $\chi_{39}(17,·)$, $\chi_{39}(22,·)$, $\chi_{39}(23,·)$, $\chi_{39}(25,·)$, $\chi_{39}(29,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2146975} a^{11} - \frac{601079}{2146975} a^{10} + \frac{263193}{2146975} a^{9} + \frac{330818}{2146975} a^{8} - \frac{1898}{3925} a^{7} + \frac{885582}{2146975} a^{6} - \frac{10331}{429395} a^{5} - \frac{173673}{429395} a^{4} - \frac{288687}{2146975} a^{3} + \frac{591126}{2146975} a^{2} + \frac{793812}{2146975} a + \frac{794667}{2146975}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{528784}{2146975} a^{11} + \frac{631961}{2146975} a^{10} - \frac{3180837}{2146975} a^{9} + \frac{2110713}{2146975} a^{8} - \frac{21868}{3925} a^{7} + \frac{8006812}{2146975} a^{6} - \frac{4189636}{429395} a^{5} + \frac{1224377}{429395} a^{4} - \frac{20472617}{2146975} a^{3} + \frac{10854341}{2146975} a^{2} - \frac{6443283}{2146975} a + \frac{899047}{2146975} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 120.784031363 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{13})\), 6.0.771147.1, 6.0.10024911.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$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}$