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Properties

Label 12.12.6836355090...2097.1
Degree $12$
Signature $[12, 0]$
Discriminant $7^{8}\cdot 17^{9}$
Root discriminant $30.64$
Ramified primes $7, 17$
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![13, 288, -388, -785, 925, 716, -742, -249, 232, 31, -28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*x + 13)
 
gp: K = bnfinit(x^12 - x^11 - 28*x^10 + 31*x^9 + 232*x^8 - 249*x^7 - 742*x^6 + 716*x^5 + 925*x^4 - 785*x^3 - 388*x^2 + 288*x + 13, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 28 x^{10} + 31 x^{9} + 232 x^{8} - 249 x^{7} - 742 x^{6} + 716 x^{5} + 925 x^{4} - 785 x^{3} - 388 x^{2} + 288 x + 13 \)

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:  \(683635509017782097=7^{8}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.64$
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:  $7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(64,·)$, $\chi_{119}(1,·)$, $\chi_{119}(67,·)$, $\chi_{119}(4,·)$, $\chi_{119}(72,·)$, $\chi_{119}(106,·)$, $\chi_{119}(18,·)$, $\chi_{119}(16,·)$, $\chi_{119}(81,·)$, $\chi_{119}(50,·)$, $\chi_{119}(86,·)$, $\chi_{119}(30,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{10} + \frac{5}{26} a^{9} + \frac{3}{26} a^{8} - \frac{11}{26} a^{7} - \frac{1}{2} a^{5} - \frac{1}{26} a^{4} - \frac{5}{26} a^{3} - \frac{3}{26} a^{2} + \frac{11}{26} a$, $\frac{1}{1257508226} a^{11} - \frac{223763}{96731402} a^{10} - \frac{14378841}{1257508226} a^{9} + \frac{1362805}{96731402} a^{8} - \frac{29079848}{628754113} a^{7} + \frac{23510455}{96731402} a^{6} + \frac{447519435}{1257508226} a^{5} - \frac{23296539}{96731402} a^{4} - \frac{282073237}{1257508226} a^{3} - \frac{27729885}{96731402} a^{2} - \frac{31149152}{628754113} a - \frac{513474}{48365701}$

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:  \( 122189.376313 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$17$17.12.9.1$x^{12} - 34 x^{8} - 10115 x^{4} - 397953$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$