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Properties

Label 12.0.213838914125824.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 13^{8}$
Root discriminant $15.64$
Ramified primes $2, 13$
Class number $1$
Class group Trivial
Galois group $A_4$ (as 12T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47, -156, 212, -228, 165, -50, 110, -26, 18, -12, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 12*x^9 + 18*x^8 - 26*x^7 + 110*x^6 - 50*x^5 + 165*x^4 - 228*x^3 + 212*x^2 - 156*x + 47)
 
gp: K = bnfinit(x^12 - 2*x^11 - 12*x^9 + 18*x^8 - 26*x^7 + 110*x^6 - 50*x^5 + 165*x^4 - 228*x^3 + 212*x^2 - 156*x + 47, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 12 x^{9} + 18 x^{8} - 26 x^{7} + 110 x^{6} - 50 x^{5} + 165 x^{4} - 228 x^{3} + 212 x^{2} - 156 x + 47 \)

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:  \(213838914125824=2^{18}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.64$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{159} a^{10} - \frac{19}{159} a^{9} - \frac{5}{53} a^{8} + \frac{40}{159} a^{7} - \frac{44}{159} a^{6} - \frac{26}{53} a^{5} - \frac{16}{159} a^{4} - \frac{20}{159} a^{3} - \frac{23}{159} a^{2} - \frac{20}{159} a + \frac{5}{159}$, $\frac{1}{5857598955} a^{11} + \frac{2863999}{1171519791} a^{10} + \frac{186966635}{1171519791} a^{9} - \frac{2613407072}{5857598955} a^{8} + \frac{2887763719}{5857598955} a^{7} - \frac{171844178}{5857598955} a^{6} + \frac{1995386194}{5857598955} a^{5} - \frac{519040209}{1952532985} a^{4} + \frac{2884720766}{5857598955} a^{3} + \frac{927137673}{1952532985} a^{2} + \frac{308522659}{1171519791} a - \frac{12977233}{124629765}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 167.105330062 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4$ (as 12T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 4 conjugacy class representatives for $A_4$
Character table for $A_4$

Intermediate fields

3.3.169.1, 4.0.10816.1 x4, 6.2.1827904.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 4 sibling: data not computed
Degree 6 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$