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Properties

Label 12.12.1591282307...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{20}\cdot 3^{16}\cdot 5^{10}\cdot 19^{2}$
Root discriminant $85.80$
Ramified primes $2, 3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 12T215

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1147904, -58368, 2582784, 2075392, 231312, -258912, -68656, 10416, 4284, -136, -108, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 108*x^10 - 136*x^9 + 4284*x^8 + 10416*x^7 - 68656*x^6 - 258912*x^5 + 231312*x^4 + 2075392*x^3 + 2582784*x^2 - 58368*x - 1147904)
 
gp: K = bnfinit(x^12 - 108*x^10 - 136*x^9 + 4284*x^8 + 10416*x^7 - 68656*x^6 - 258912*x^5 + 231312*x^4 + 2075392*x^3 + 2582784*x^2 - 58368*x - 1147904, 1)
 

Normalized defining polynomial

\( x^{12} - 108 x^{10} - 136 x^{9} + 4284 x^{8} + 10416 x^{7} - 68656 x^{6} - 258912 x^{5} + 231312 x^{4} + 2075392 x^{3} + 2582784 x^{2} - 58368 x - 1147904 \)

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:  \(159128230717440000000000=2^{20}\cdot 3^{16}\cdot 5^{10}\cdot 19^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.80$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{6} - \frac{1}{8} a^{4} - \frac{1}{6} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{48} a^{7} + \frac{1}{12} a^{4} + \frac{1}{6} a$, $\frac{1}{96} a^{8} + \frac{1}{24} a^{5} - \frac{1}{4} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{96} a^{9} - \frac{1}{12} a^{3} - \frac{1}{3}$, $\frac{1}{768} a^{10} - \frac{1}{192} a^{9} - \frac{1}{192} a^{8} - \frac{1}{96} a^{7} - \frac{1}{192} a^{6} + \frac{1}{24} a^{5} - \frac{1}{48} a^{4} + \frac{5}{24} a^{3} + \frac{1}{48} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{989693584896} a^{11} + \frac{63000013}{247423396224} a^{10} + \frac{570158759}{247423396224} a^{9} + \frac{53372971}{123711698112} a^{8} - \frac{2105073161}{247423396224} a^{7} - \frac{127292281}{30927924528} a^{6} - \frac{649232515}{20618616352} a^{5} - \frac{1942130897}{30927924528} a^{4} + \frac{1040261299}{20618616352} a^{3} + \frac{318881183}{15463962264} a^{2} - \frac{309221957}{2577327044} a + \frac{960986939}{1932995283}$

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:  \( 103895287.488 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T215:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 18 conjugacy class representatives for 1/2[F_36^2]2
Character table for 1/2[F_36^2]2

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\)

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 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 R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
$3$3.12.16.8$x^{12} + 90 x^{11} + 315 x^{10} - 84 x^{9} - 225 x^{8} - 243 x^{7} - 9 x^{6} + 54 x^{5} - 243 x^{4} - 135 x^{3} - 162 x^{2} + 243 x - 162$$3$$4$$16$12T173$[2, 2, 2, 2]^{8}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$