Properties

Label 12.8.17188851012...752.17
Degree $12$
Signature $[8, 2]$
Discriminant $2^{21}\cdot 31^{10}$
Root discriminant $58.83$
Ramified primes $2, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 12T188

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 24*x^10 + 65*x^9 + 88*x^8 - 784*x^7 + 800*x^6 + 3123*x^5 - 2443*x^4 + 1956*x^3 + 1756*x^2 - 1498*x + 94)
 
gp: K = bnfinit(x^12 - 2*x^11 - 24*x^10 + 65*x^9 + 88*x^8 - 784*x^7 + 800*x^6 + 3123*x^5 - 2443*x^4 + 1956*x^3 + 1756*x^2 - 1498*x + 94, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94, -1498, 1756, 1956, -2443, 3123, 800, -784, 88, 65, -24, -2, 1]);
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 24 x^{10} + 65 x^{9} + 88 x^{8} - 784 x^{7} + 800 x^{6} + 3123 x^{5} - 2443 x^{4} + 1956 x^{3} + 1756 x^{2} - 1498 x + 94 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1718885101298360778752=2^{21}\cdot 31^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $58.83$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{31} a^{6} - \frac{1}{31} a^{5} + \frac{3}{31} a^{4} - \frac{11}{31} a^{3} + \frac{13}{31} a^{2} - \frac{5}{31} a + \frac{1}{31}$, $\frac{1}{31} a^{7} + \frac{2}{31} a^{5} - \frac{8}{31} a^{4} + \frac{2}{31} a^{3} + \frac{8}{31} a^{2} - \frac{4}{31} a + \frac{1}{31}$, $\frac{1}{31} a^{8} - \frac{6}{31} a^{5} - \frac{4}{31} a^{4} - \frac{1}{31} a^{3} + \frac{1}{31} a^{2} + \frac{11}{31} a - \frac{2}{31}$, $\frac{1}{31} a^{9} - \frac{10}{31} a^{5} - \frac{14}{31} a^{4} - \frac{3}{31} a^{3} - \frac{4}{31} a^{2} - \frac{1}{31} a + \frac{6}{31}$, $\frac{1}{186} a^{10} - \frac{1}{186} a^{8} - \frac{1}{62} a^{7} - \frac{1}{186} a^{6} + \frac{13}{62} a^{5} + \frac{7}{62} a^{4} + \frac{13}{31} a^{3} - \frac{32}{93} a^{2} - \frac{19}{93} a + \frac{35}{93}$, $\frac{1}{26296175196} a^{11} - \frac{52437421}{26296175196} a^{10} + \frac{129634445}{26296175196} a^{9} - \frac{197732131}{13148087598} a^{8} + \frac{10613336}{6574043799} a^{7} + \frac{32363843}{13148087598} a^{6} - \frac{113656107}{730449311} a^{5} - \frac{1306792679}{2921797244} a^{4} - \frac{3104669575}{6574043799} a^{3} - \frac{3135688810}{6574043799} a^{2} + \frac{33507793}{730449311} a - \frac{5296400075}{13148087598}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3358183.91941 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

12T188:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 768
The 32 conjugacy class representatives for [2^6]A_4=2wrA_4(6)
Character table for [2^6]A_4=2wrA_4(6) is not computed

Intermediate fields

3.3.961.1, 6.6.59105344.1

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 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.11.13$x^{4} + 4 x^{2} + 14$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
2.4.10.7$x^{4} - 2 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$