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Properties

Label 12.12.2714751839...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{16}\cdot 5^{8}\cdot 13^{9}$
Root discriminant $50.44$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1093, 2126, -7526, -12222, 7015, 10132, -2964, -2348, 515, 206, -38, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 38*x^10 + 206*x^9 + 515*x^8 - 2348*x^7 - 2964*x^6 + 10132*x^5 + 7015*x^4 - 12222*x^3 - 7526*x^2 + 2126*x + 1093)
 
gp: K = bnfinit(x^12 - 6*x^11 - 38*x^10 + 206*x^9 + 515*x^8 - 2348*x^7 - 2964*x^6 + 10132*x^5 + 7015*x^4 - 12222*x^3 - 7526*x^2 + 2126*x + 1093, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} - 38 x^{10} + 206 x^{9} + 515 x^{8} - 2348 x^{7} - 2964 x^{6} + 10132 x^{5} + 7015 x^{4} - 12222 x^{3} - 7526 x^{2} + 2126 x + 1093 \)

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:  \(271475183948800000000=2^{16}\cdot 5^{8}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.44$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{5176} a^{9} + \frac{225}{5176} a^{8} - \frac{123}{2588} a^{7} + \frac{227}{2588} a^{6} + \frac{269}{1294} a^{5} + \frac{25}{1294} a^{4} + \frac{257}{2588} a^{3} + \frac{343}{2588} a^{2} + \frac{2103}{5176} a - \frac{313}{5176}$, $\frac{1}{5176} a^{10} + \frac{121}{2588} a^{8} + \frac{81}{2588} a^{7} - \frac{71}{2588} a^{6} - \frac{11}{2588} a^{5} + \frac{3}{1294} a^{4} + \frac{101}{2588} a^{3} + \frac{445}{5176} a^{2} + \frac{705}{2588} a + \frac{1245}{2588}$, $\frac{1}{1681133744} a^{11} - \frac{129475}{1681133744} a^{10} + \frac{55603}{1681133744} a^{9} + \frac{94384669}{1681133744} a^{8} + \frac{100185311}{840566872} a^{7} + \frac{58092593}{840566872} a^{6} - \frac{10294013}{840566872} a^{5} + \frac{136835159}{840566872} a^{4} + \frac{46008193}{1681133744} a^{3} + \frac{373676569}{1681133744} a^{2} - \frac{489413417}{1681133744} a + \frac{703582285}{1681133744}$

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:  \( 1507027.09552 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.1300.1 x3, 4.4.35152.1, 6.6.21970000.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$2$2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$5$5.12.8.1$x^{12} - 30 x^{9} + 175 x^{6} + 500 x^{3} + 5000$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$