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Properties

Label 12.12.5514297181...3041.1
Degree $12$
Signature $[12, 0]$
Discriminant $3^{16}\cdot 71^{6}$
Root discriminant $36.46$
Ramified primes $3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![648, 324, -4212, -1899, 7236, 2880, -3153, -882, 543, 92, -39, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648)
 
gp: K = bnfinit(x^12 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} - 39 x^{10} + 92 x^{9} + 543 x^{8} - 882 x^{7} - 3153 x^{6} + 2880 x^{5} + 7236 x^{4} - 1899 x^{3} - 4212 x^{2} + 324 x + 648 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5514297181968073041=3^{16}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.46$
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:  $3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{8} - \frac{1}{24} a^{7} + \frac{1}{24} a^{6} + \frac{7}{36} a^{5} + \frac{1}{6} a^{4} + \frac{7}{24} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{432} a^{9} + \frac{1}{72} a^{7} - \frac{13}{432} a^{6} - \frac{1}{8} a^{5} + \frac{1}{48} a^{4} + \frac{49}{144} a^{3} - \frac{1}{12} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1728} a^{10} + \frac{1}{1728} a^{9} - \frac{1}{288} a^{8} + \frac{65}{1728} a^{7} - \frac{139}{1728} a^{6} - \frac{107}{576} a^{5} - \frac{47}{144} a^{4} - \frac{143}{576} a^{3} - \frac{5}{48} a^{2} - \frac{7}{16} a + \frac{1}{8}$, $\frac{1}{5609088} a^{11} + \frac{79}{467424} a^{10} + \frac{1871}{1869696} a^{9} + \frac{27599}{5609088} a^{8} + \frac{6223}{233712} a^{7} + \frac{15919}{311616} a^{6} + \frac{338611}{1869696} a^{5} + \frac{30919}{69248} a^{4} + \frac{59471}{207744} a^{3} + \frac{3065}{12984} a^{2} + \frac{4191}{17312} a - \frac{223}{8656}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 1188211.01781 \) (assuming GRH)
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

\(\Q(\zeta_{9})^+\), 4.4.408321.1 x4, 6.6.33074001.1 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$71$71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$