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Properties

Label 12.0.456488925854205933.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 13^{9}$
Root discriminant $29.62$
Ramified primes $3, 13$
Class number $9$
Class group $[3, 3]$
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![1569, 567, -822, -286, 348, -60, 140, 30, 3, 25, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 3*x^10 + 25*x^9 + 3*x^8 + 30*x^7 + 140*x^6 - 60*x^5 + 348*x^4 - 286*x^3 - 822*x^2 + 567*x + 1569)
 
gp: K = bnfinit(x^12 - 3*x^11 - 3*x^10 + 25*x^9 + 3*x^8 + 30*x^7 + 140*x^6 - 60*x^5 + 348*x^4 - 286*x^3 - 822*x^2 + 567*x + 1569, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} - 3 x^{10} + 25 x^{9} + 3 x^{8} + 30 x^{7} + 140 x^{6} - 60 x^{5} + 348 x^{4} - 286 x^{3} - 822 x^{2} + 567 x + 1569 \)

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:  \(456488925854205933=3^{16}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.62$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(64,·)$, $\chi_{117}(1,·)$, $\chi_{117}(34,·)$, $\chi_{117}(70,·)$, $\chi_{117}(103,·)$, $\chi_{117}(40,·)$, $\chi_{117}(73,·)$, $\chi_{117}(109,·)$, $\chi_{117}(79,·)$, $\chi_{117}(112,·)$, $\chi_{117}(25,·)$, $\chi_{117}(31,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{44531070717242313} a^{11} + \frac{1442529418467692}{44531070717242313} a^{10} - \frac{1365778327052417}{44531070717242313} a^{9} + \frac{2142269122906795}{4947896746360257} a^{8} - \frac{5326208543129435}{14843690239080771} a^{7} - \frac{353568744165969}{1649298915453419} a^{6} - \frac{19195811982746248}{44531070717242313} a^{5} - \frac{5431556534653790}{44531070717242313} a^{4} - \frac{17079360723144139}{44531070717242313} a^{3} + \frac{441064195084180}{4947896746360257} a^{2} + \frac{3899485709527817}{14843690239080771} a + \frac{825761160996343}{14843690239080771}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1744.53950676 \)
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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 4.0.2197.1, 6.6.14414517.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$13$13.12.9.2$x^{12} - 52 x^{8} + 676 x^{4} - 79092$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$