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Properties

Label 12.0.24016663313...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{6}\cdot 7^{6}\cdot 13^{11}$
Root discriminant $107.57$
Ramified primes $3, 5, 7, 13$
Class number $101632$ (GRH)
Class group $[2, 2, 2, 2, 4, 1588]$ (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![5183066071, -1167160983, 1167160983, -85955767, 85955767, -2786135, 2786135, -44279, 44279, -339, 339, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 339*x^10 - 339*x^9 + 44279*x^8 - 44279*x^7 + 2786135*x^6 - 2786135*x^5 + 85955767*x^4 - 85955767*x^3 + 1167160983*x^2 - 1167160983*x + 5183066071)
 
gp: K = bnfinit(x^12 - x^11 + 339*x^10 - 339*x^9 + 44279*x^8 - 44279*x^7 + 2786135*x^6 - 2786135*x^5 + 85955767*x^4 - 85955767*x^3 + 1167160983*x^2 - 1167160983*x + 5183066071, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 339 x^{10} - 339 x^{9} + 44279 x^{8} - 44279 x^{7} + 2786135 x^{6} - 2786135 x^{5} + 85955767 x^{4} - 85955767 x^{3} + 1167160983 x^{2} - 1167160983 x + 5183066071 \)

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:  \(2401666331349765944953125=3^{6}\cdot 5^{6}\cdot 7^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.57$
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:  $3, 5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1365=3\cdot 5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1365}(1,·)$, $\chi_{1365}(1156,·)$, $\chi_{1365}(839,·)$, $\chi_{1365}(841,·)$, $\chi_{1365}(1259,·)$, $\chi_{1365}(944,·)$, $\chi_{1365}(946,·)$, $\chi_{1365}(211,·)$, $\chi_{1365}(629,·)$, $\chi_{1365}(314,·)$, $\chi_{1365}(316,·)$, $\chi_{1365}(734,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{591920083} a^{7} + \frac{277505889}{591920083} a^{6} + \frac{182}{591920083} a^{5} + \frac{80752625}{591920083} a^{4} + \frac{9464}{591920083} a^{3} + \frac{189751960}{591920083} a^{2} + \frac{123032}{591920083} a + \frac{44042288}{591920083}$, $\frac{1}{591920083} a^{8} + \frac{208}{591920083} a^{6} - \frac{112112118}{591920083} a^{5} + \frac{13520}{591920083} a^{4} + \frac{223426735}{591920083} a^{3} + \frac{281216}{591920083} a^{2} - \frac{110105720}{591920083} a + \frac{913952}{591920083}$, $\frac{1}{591920083} a^{9} + \frac{174831104}{591920083} a^{6} - \frac{24336}{591920083} a^{5} + \frac{643059}{591920083} a^{4} - \frac{1687296}{591920083} a^{3} + \frac{80132161}{591920083} a^{2} - \frac{24676704}{591920083} a - \frac{281994659}{591920083}$, $\frac{1}{591920083} a^{10} - \frac{30420}{591920083} a^{6} + \frac{145066613}{591920083} a^{5} - \frac{2636400}{591920083} a^{4} - \frac{104804110}{591920083} a^{3} - \frac{61691760}{591920083} a^{2} + \frac{273434233}{591920083} a - \frac{213864768}{591920083}$, $\frac{1}{591920083} a^{11} - \frac{90013753}{591920083} a^{6} + \frac{2900040}{591920083} a^{5} - \frac{78296060}{591920083} a^{4} + \frac{226203120}{591920083} a^{3} + \frac{123408017}{591920083} a^{2} - \frac{22751826}{591920083} a + \frac{251253131}{591920083}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{1588}$, which has order $101632$ (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:  \( 120.784031363 \) (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, 4.0.24221925.2, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R R ${\href{/LocalNumberField/11.12.0.1}{12} }$ 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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$7$7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$