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Properties

Label 12.12.3174921459...1757.1
Degree $12$
Signature $[12, 0]$
Discriminant $11^{6}\cdot 13^{11}$
Root discriminant $34.82$
Ramified primes $11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1559, 11036, -11036, -11077, 11077, 3665, -3665, -547, 547, 38, -38, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 38*x^10 + 38*x^9 + 547*x^8 - 547*x^7 - 3665*x^6 + 3665*x^5 + 11077*x^4 - 11077*x^3 - 11036*x^2 + 11036*x - 1559)
 
gp: K = bnfinit(x^12 - x^11 - 38*x^10 + 38*x^9 + 547*x^8 - 547*x^7 - 3665*x^6 + 3665*x^5 + 11077*x^4 - 11077*x^3 - 11036*x^2 + 11036*x - 1559, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 38 x^{10} + 38 x^{9} + 547 x^{8} - 547 x^{7} - 3665 x^{6} + 3665 x^{5} + 11077 x^{4} - 11077 x^{3} - 11036 x^{2} + 11036 x - 1559 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3174921459820581757=11^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.82$
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:  $11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(32,·)$, $\chi_{143}(1,·)$, $\chi_{143}(98,·)$, $\chi_{143}(100,·)$, $\chi_{143}(133,·)$, $\chi_{143}(12,·)$, $\chi_{143}(76,·)$, $\chi_{143}(109,·)$, $\chi_{143}(21,·)$, $\chi_{143}(54,·)$, $\chi_{143}(23,·)$, $\chi_{143}(56,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{599} a^{7} - \frac{188}{599} a^{6} - \frac{21}{599} a^{5} - \frac{210}{599} a^{4} + \frac{126}{599} a^{3} - \frac{253}{599} a^{2} - \frac{189}{599} a - \frac{31}{599}$, $\frac{1}{599} a^{8} - \frac{24}{599} a^{6} + \frac{35}{599} a^{5} + \frac{180}{599} a^{4} + \frac{74}{599} a^{3} + \frac{167}{599} a^{2} - \frac{222}{599} a + \frac{162}{599}$, $\frac{1}{599} a^{9} - \frac{284}{599} a^{6} + \frac{275}{599} a^{5} - \frac{174}{599} a^{4} + \frac{196}{599} a^{3} + \frac{295}{599} a^{2} - \frac{181}{599} a - \frac{145}{599}$, $\frac{1}{599} a^{10} + \frac{194}{599} a^{6} - \frac{148}{599} a^{5} - \frac{143}{599} a^{4} + \frac{139}{599} a^{3} - \frac{153}{599} a^{2} + \frac{89}{599} a + \frac{181}{599}$, $\frac{1}{599} a^{11} - \frac{215}{599} a^{6} - \frac{262}{599} a^{5} + \frac{147}{599} a^{4} - \frac{38}{599} a^{3} + \frac{53}{599} a^{2} - \frac{291}{599} a + \frac{24}{599}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 162467.601473 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.4.265837.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ R R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$11$11.12.6.2$x^{12} + 14641 x^{4} - 322102 x^{2} + 14172488$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$