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Properties

Label 12.0.42516909149...4144.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 7^{6}\cdot 13^{10}$
Root discriminant $63.44$
Ramified primes $2, 7, 13$
Class number $4536$ (GRH)
Class group $[3, 3, 3, 168]$ (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![15418831, -1713202, 4767669, -455716, 664294, -53204, 53384, -3418, 2619, -122, 75, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 75*x^10 - 122*x^9 + 2619*x^8 - 3418*x^7 + 53384*x^6 - 53204*x^5 + 664294*x^4 - 455716*x^3 + 4767669*x^2 - 1713202*x + 15418831)
 
gp: K = bnfinit(x^12 - 2*x^11 + 75*x^10 - 122*x^9 + 2619*x^8 - 3418*x^7 + 53384*x^6 - 53204*x^5 + 664294*x^4 - 455716*x^3 + 4767669*x^2 - 1713202*x + 15418831, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 75 x^{10} - 122 x^{9} + 2619 x^{8} - 3418 x^{7} + 53384 x^{6} - 53204 x^{5} + 664294 x^{4} - 455716 x^{3} + 4767669 x^{2} - 1713202 x + 15418831 \)

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:  \(4251690914950152454144=2^{18}\cdot 7^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.44$
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$$\chi_{728}(1,·)$, $\chi_{728}(69,·)$, $\chi_{728}(225,·)$, $\chi_{728}(393,·)$, $\chi_{728}(237,·)$, $\chi_{728}(337,·)$, $\chi_{728}(685,·)$, $\chi_{728}(113,·)$, $\chi_{728}(181,·)$, $\chi_{728}(673,·)$, $\chi_{728}(573,·)$, $\chi_{728}(517,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{15} a^{10} + \frac{2}{15} a^{9} - \frac{4}{15} a^{8} - \frac{7}{15} a^{7} + \frac{4}{15} a^{6} + \frac{2}{15} a^{5} - \frac{2}{5} a^{4} - \frac{2}{15} a^{3} + \frac{1}{5} a^{2} - \frac{1}{3} a - \frac{2}{15}$, $\frac{1}{10982489237237139611328705} a^{11} + \frac{68212288104714263967752}{2196497847447427922265741} a^{10} + \frac{241308834123499978423579}{3660829745745713203776235} a^{9} + \frac{1140501584388647870533112}{3660829745745713203776235} a^{8} + \frac{3191411786128018178348678}{10982489237237139611328705} a^{7} - \frac{1087685613721603949972336}{10982489237237139611328705} a^{6} + \frac{116869681042754662045934}{2196497847447427922265741} a^{5} + \frac{274058348708239618041815}{2196497847447427922265741} a^{4} + \frac{3844955799481417199349317}{10982489237237139611328705} a^{3} + \frac{4611059538211017814887454}{10982489237237139611328705} a^{2} + \frac{142702592275675383747226}{3660829745745713203776235} a - \frac{20437687571684852516111}{83835795704100302376555}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{168}$, which has order $4536$ (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

\(\Q(\sqrt{-182}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-14})\), 6.0.65204991488.6, 6.0.5015768576.6, \(\Q(\zeta_{13})^+\)

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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$7$7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$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}$