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Properties

Label 12.12.5163100639...8576.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{42}\cdot 3^{12}\cdot 47^{2}$
Root discriminant $64.48$
Ramified primes $2, 3, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 12T173

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![752, 4512, 10152, 9480, 783, -4536, -2068, 600, 450, -24, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 36*x^10 - 24*x^9 + 450*x^8 + 600*x^7 - 2068*x^6 - 4536*x^5 + 783*x^4 + 9480*x^3 + 10152*x^2 + 4512*x + 752)
 
gp: K = bnfinit(x^12 - 36*x^10 - 24*x^9 + 450*x^8 + 600*x^7 - 2068*x^6 - 4536*x^5 + 783*x^4 + 9480*x^3 + 10152*x^2 + 4512*x + 752, 1)
 

Normalized defining polynomial

\( x^{12} - 36 x^{10} - 24 x^{9} + 450 x^{8} + 600 x^{7} - 2068 x^{6} - 4536 x^{5} + 783 x^{4} + 9480 x^{3} + 10152 x^{2} + 4512 x + 752 \)

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:  \(5163100639119934488576=2^{42}\cdot 3^{12}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.48$
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:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{1}{32} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} + \frac{31}{64} a^{2} - \frac{5}{16} a - \frac{5}{16}$, $\frac{1}{512} a^{11} + \frac{1}{256} a^{10} - \frac{1}{16} a^{9} - \frac{11}{64} a^{8} - \frac{119}{256} a^{7} + \frac{31}{128} a^{6} + \frac{57}{128} a^{5} + \frac{1}{32} a^{4} - \frac{209}{512} a^{3} - \frac{77}{256} a^{2} + \frac{29}{128} a + \frac{17}{64}$

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:  \( 24309193.929 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T173:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 18 conjugacy class representatives for [3^4:2]4_4
Character table for [3^4:2]4_4

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\)

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: 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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ R ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
$3$3.12.12.13$x^{12} + 42 x^{11} - 57 x^{10} - 36 x^{9} - 117 x^{8} - 54 x^{7} - 18 x^{6} - 27 x^{5} + 54 x^{4} + 108 x^{3} - 81 x^{2} + 81 x + 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.3.2.1$x^{3} - 47$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$