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Properties

Label 12.12.4108400332...3397.1
Degree $12$
Signature $[12, 0]$
Discriminant $3^{18}\cdot 13^{9}$
Root discriminant $35.57$
Ramified primes $3, 13$
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![-51, -243, -72, 858, 600, -909, -752, 285, 279, -17, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 30*x^10 - 17*x^9 + 279*x^8 + 285*x^7 - 752*x^6 - 909*x^5 + 600*x^4 + 858*x^3 - 72*x^2 - 243*x - 51)
 
gp: K = bnfinit(x^12 - 30*x^10 - 17*x^9 + 279*x^8 + 285*x^7 - 752*x^6 - 909*x^5 + 600*x^4 + 858*x^3 - 72*x^2 - 243*x - 51, 1)
 

Normalized defining polynomial

\( x^{12} - 30 x^{10} - 17 x^{9} + 279 x^{8} + 285 x^{7} - 752 x^{6} - 909 x^{5} + 600 x^{4} + 858 x^{3} - 72 x^{2} - 243 x - 51 \)

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:  \(4108400332687853397=3^{18}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.57$
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$$\chi_{117}(64,·)$, $\chi_{117}(1,·)$, $\chi_{117}(5,·)$, $\chi_{117}(103,·)$, $\chi_{117}(8,·)$, $\chi_{117}(44,·)$, $\chi_{117}(47,·)$, $\chi_{117}(40,·)$, $\chi_{117}(83,·)$, $\chi_{117}(86,·)$, $\chi_{117}(25,·)$, $\chi_{117}(79,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} + \frac{8}{17} a^{9} + \frac{3}{17} a^{8} - \frac{3}{17} a^{7} - \frac{8}{17} a^{6} + \frac{8}{17} a^{5} + \frac{2}{17} a^{4} - \frac{2}{17} a^{3} - \frac{5}{17} a^{2} - \frac{4}{17} a$, $\frac{1}{505630711} a^{11} + \frac{13980801}{505630711} a^{10} - \frac{57069019}{505630711} a^{9} + \frac{89992094}{505630711} a^{8} + \frac{124232579}{505630711} a^{7} + \frac{175592809}{505630711} a^{6} - \frac{85002672}{505630711} a^{5} + \frac{249711641}{505630711} a^{4} + \frac{203102154}{505630711} a^{3} + \frac{95213988}{505630711} a^{2} - \frac{205608784}{505630711} a + \frac{1647686}{29742983}$

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:  \( 520561.482962 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 4.4.19773.1, 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 ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$13$13.12.9.2$x^{12} - 52 x^{8} + 676 x^{4} - 79092$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$