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Properties

Label 12.8.35291480629...0208.2
Degree $12$
Signature $[8, 2]$
Discriminant $2^{23}\cdot 29^{10}$
Root discriminant $62.47$
Ramified primes $2, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $A_4:C_4$ (as 12T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-328, -952, 2078, 4624, -5547, -3024, 4494, -884, -358, 168, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 12*x^10 + 168*x^9 - 358*x^8 - 884*x^7 + 4494*x^6 - 3024*x^5 - 5547*x^4 + 4624*x^3 + 2078*x^2 - 952*x - 328)
 
gp: K = bnfinit(x^12 - 4*x^11 - 12*x^10 + 168*x^9 - 358*x^8 - 884*x^7 + 4494*x^6 - 3024*x^5 - 5547*x^4 + 4624*x^3 + 2078*x^2 - 952*x - 328, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 12 x^{10} + 168 x^{9} - 358 x^{8} - 884 x^{7} + 4494 x^{6} - 3024 x^{5} - 5547 x^{4} + 4624 x^{3} + 2078 x^{2} - 952 x - 328 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3529148062919932510208=2^{23}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{58} a^{6} - \frac{1}{29} a^{5} - \frac{4}{29} a^{4} + \frac{5}{29} a^{3} - \frac{17}{58} a^{2} + \frac{5}{29} a + \frac{11}{29}$, $\frac{1}{58} a^{7} - \frac{6}{29} a^{5} - \frac{3}{29} a^{4} + \frac{3}{58} a^{3} - \frac{12}{29} a^{2} - \frac{8}{29} a - \frac{7}{29}$, $\frac{1}{58} a^{8} + \frac{14}{29} a^{5} + \frac{23}{58} a^{4} - \frac{10}{29} a^{3} + \frac{6}{29} a^{2} - \frac{5}{29} a - \frac{13}{29}$, $\frac{1}{116} a^{9} - \frac{1}{116} a^{7} - \frac{25}{116} a^{5} + \frac{9}{29} a^{4} + \frac{19}{116} a^{3} - \frac{8}{29} a^{2} - \frac{1}{2} a + \frac{9}{29}$, $\frac{1}{580} a^{10} + \frac{1}{290} a^{9} - \frac{3}{580} a^{8} + \frac{1}{145} a^{7} - \frac{1}{116} a^{6} - \frac{33}{290} a^{5} - \frac{267}{580} a^{4} + \frac{8}{145} a^{3} + \frac{33}{290} a^{2} + \frac{11}{145} a - \frac{54}{145}$, $\frac{1}{798056800} a^{11} + \frac{520213}{798056800} a^{10} - \frac{2918511}{798056800} a^{9} - \frac{2757359}{798056800} a^{8} - \frac{5990701}{798056800} a^{7} + \frac{5091319}{798056800} a^{6} - \frac{214856883}{798056800} a^{5} - \frac{79301903}{159611360} a^{4} + \frac{22794169}{399028400} a^{3} + \frac{3780253}{79805680} a^{2} + \frac{2360174}{24939275} a - \frac{537867}{2433100}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 36152561.893 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 12T30):

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

\(\Q(\sqrt{2}) \), 3.3.6728.1 x3, 6.6.362127872.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
$29$29.12.10.3$x^{12} + 232 x^{6} + 22707$$6$$2$$10$$C_3 : C_4$$[\ ]_{6}^{2}$