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Properties

Label 12.0.22586547602...312.17
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![1191311, -2274088, 1741556, -693812, 165147, -26688, 2232, 856, -619, 168, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 12*x^10 + 168*x^9 - 619*x^8 + 856*x^7 + 2232*x^6 - 26688*x^5 + 165147*x^4 - 693812*x^3 + 1741556*x^2 - 2274088*x + 1191311)
 
gp: K = bnfinit(x^12 - 4*x^11 - 12*x^10 + 168*x^9 - 619*x^8 + 856*x^7 + 2232*x^6 - 26688*x^5 + 165147*x^4 - 693812*x^3 + 1741556*x^2 - 2274088*x + 1191311, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 12 x^{10} + 168 x^{9} - 619 x^{8} + 856 x^{7} + 2232 x^{6} - 26688 x^{5} + 165147 x^{4} - 693812 x^{3} + 1741556 x^{2} - 2274088 x + 1191311 \)

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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{116} a^{6} - \frac{1}{58} a^{5} + \frac{21}{116} a^{4} + \frac{5}{58} a^{3} + \frac{41}{116} a^{2} - \frac{12}{29} a - \frac{7}{116}$, $\frac{1}{116} a^{7} + \frac{17}{116} a^{5} - \frac{3}{58} a^{4} + \frac{3}{116} a^{3} - \frac{6}{29} a^{2} - \frac{45}{116} a - \frac{7}{58}$, $\frac{1}{232} a^{8} + \frac{7}{58} a^{5} - \frac{3}{116} a^{4} - \frac{5}{58} a^{3} + \frac{3}{58} a^{2} + \frac{6}{29} a - \frac{55}{232}$, $\frac{1}{232} a^{9} + \frac{25}{116} a^{5} - \frac{7}{58} a^{4} - \frac{9}{58} a^{3} + \frac{15}{58} a^{2} - \frac{103}{232} a + \frac{10}{29}$, $\frac{1}{464} a^{10} - \frac{1}{464} a^{8} - \frac{1}{232} a^{6} + \frac{3}{29} a^{5} - \frac{39}{232} a^{4} + \frac{3}{58} a^{3} - \frac{159}{464} a^{2} - \frac{3}{58} a - \frac{161}{464}$, $\frac{1}{894870151077920} a^{11} - \frac{263052164561}{894870151077920} a^{10} + \frac{336451033757}{178974030215584} a^{9} + \frac{1745845347563}{894870151077920} a^{8} + \frac{346359378611}{89487015107792} a^{7} + \frac{997088254773}{447435075538960} a^{6} + \frac{8646746900251}{89487015107792} a^{5} + \frac{111087941730041}{447435075538960} a^{4} - \frac{147969797133847}{894870151077920} a^{3} - \frac{378407666228513}{894870151077920} a^{2} - \frac{306124067565623}{894870151077920} a + \frac{128842468267963}{894870151077920}$

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:  \( 12792098.5879 \) (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, 6.2.1448511488.7

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.8.24.5$x^{8} - 15$$8$$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}$