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Properties

Label 12.0.22586547602...312.27
Degree $12$
Signature $[0, 6]$
Discriminant $2^{29}\cdot 29^{10}$
Root discriminant $88.34$
Ramified primes $2, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $A_4:C_4$ (as 12T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1956539, -298158, -62692, -133526, 983, 6772, 11800, -1916, -483, 2, 28, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 28*x^10 + 2*x^9 - 483*x^8 - 1916*x^7 + 11800*x^6 + 6772*x^5 + 983*x^4 - 133526*x^3 - 62692*x^2 - 298158*x + 1956539)
 
gp: K = bnfinit(x^12 - 6*x^11 + 28*x^10 + 2*x^9 - 483*x^8 - 1916*x^7 + 11800*x^6 + 6772*x^5 + 983*x^4 - 133526*x^3 - 62692*x^2 - 298158*x + 1956539, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 28 x^{10} + 2 x^{9} - 483 x^{8} - 1916 x^{7} + 11800 x^{6} + 6772 x^{5} + 983 x^{4} - 133526 x^{3} - 62692 x^{2} - 298158 x + 1956539 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(225865476026875680653312=2^{29}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.34$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{8} - \frac{1}{8}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{2} a^{3} - \frac{1}{16} a^{2} - \frac{1}{2} a + \frac{1}{16}$, $\frac{1}{598042290172247706163041424} a^{11} - \frac{2466568861161199077319763}{598042290172247706163041424} a^{10} + \frac{1915182702446215653197725}{299021145086123853081520712} a^{9} - \frac{1474494232902669401609209}{37377643135765481635190089} a^{8} - \frac{4390252580170690183922507}{74755286271530963270380178} a^{7} - \frac{2393025064688053369902631}{149510572543061926540760356} a^{6} + \frac{20631212564060712424097531}{149510572543061926540760356} a^{5} + \frac{10995937962519520151151903}{74755286271530963270380178} a^{4} - \frac{103041860093924292824381953}{598042290172247706163041424} a^{3} + \frac{221013026972951310201365959}{598042290172247706163041424} a^{2} - \frac{71130211666423795341997319}{299021145086123853081520712} a - \frac{11387077574411310242062451}{37377643135765481635190089}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $5$
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:  \( 5497917.25471 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 12T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

3.3.6728.1, 6.2.1448511488.4

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 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.8.24.7$x^{8} + 8 x^{7} + 12 x^{6} + 10 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$29$29.12.10.3$x^{12} + 232 x^{6} + 22707$$6$$2$$10$$C_3 : C_4$$[\ ]_{6}^{2}$