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Properties

Label 12.12.4215845235...6432.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{8}\cdot 3^{10}\cdot 19^{3}\cdot 61^{2}\cdot 103^{3}$
Root discriminant $52.33$
Ramified primes $2, 3, 19, 61, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 12T258

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![976, 5856, 13176, 12232, 693, -6372, -2898, 864, 648, -30, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 45*x^10 - 30*x^9 + 648*x^8 + 864*x^7 - 2898*x^6 - 6372*x^5 + 693*x^4 + 12232*x^3 + 13176*x^2 + 5856*x + 976)
 
gp: K = bnfinit(x^12 - 45*x^10 - 30*x^9 + 648*x^8 + 864*x^7 - 2898*x^6 - 6372*x^5 + 693*x^4 + 12232*x^3 + 13176*x^2 + 5856*x + 976, 1)
 

Normalized defining polynomial

\( x^{12} - 45 x^{10} - 30 x^{9} + 648 x^{8} + 864 x^{7} - 2898 x^{6} - 6372 x^{5} + 693 x^{4} + 12232 x^{3} + 13176 x^{2} + 5856 x + 976 \)

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:  \(421584523590712626432=2^{8}\cdot 3^{10}\cdot 19^{3}\cdot 61^{2}\cdot 103^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.33$
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:  $2, 3, 19, 61, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{10} - \frac{1}{16} a^{9} - \frac{17}{64} a^{8} - \frac{5}{32} a^{7} - \frac{7}{16} a^{6} + \frac{3}{8} a^{5} - \frac{1}{32} a^{4} + \frac{1}{16} a^{3} + \frac{13}{64} a^{2} + \frac{1}{16} a + \frac{1}{16}$, $\frac{1}{1536} a^{11} + \frac{5}{768} a^{10} + \frac{55}{1536} a^{9} + \frac{65}{192} a^{8} - \frac{37}{192} a^{7} - \frac{35}{96} a^{6} + \frac{359}{768} a^{5} - \frac{91}{192} a^{4} - \frac{443}{1536} a^{3} + \frac{61}{768} a^{2} + \frac{143}{384} a - \frac{89}{192}$

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:  \( 5455773.50531 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T258:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3888
The 39 conjugacy class representatives for [3^4:2]S(4)
Character table for [3^4:2]S(4) is not computed

Intermediate fields

4.4.1957.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.9.5$x^{9} + 3 x^{7} + 3 x^{6} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$61$61.3.2.3$x^{3} - 244$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
$103$103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$