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Properties

Label 12.0.11528772007...2752.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 397^{11}$
Root discriminant $835.25$
Ramified primes $2, 3, 397$
Class number $12193792$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 4, 4, 11908]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![74089728, 0, 493931520, 0, 826306272, 0, 39960432, 0, 403749, 0, 1191, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 1191*x^10 + 403749*x^8 + 39960432*x^6 + 826306272*x^4 + 493931520*x^2 + 74089728)
 
gp: K = bnfinit(x^12 + 1191*x^10 + 403749*x^8 + 39960432*x^6 + 826306272*x^4 + 493931520*x^2 + 74089728, 1)
 

Normalized defining polynomial

\( x^{12} + 1191 x^{10} + 403749 x^{8} + 39960432 x^{6} + 826306272 x^{4} + 493931520 x^{2} + 74089728 \)

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:  \(115287720075026486867753102672842752=2^{12}\cdot 3^{6}\cdot 397^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $835.25$
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:  $2, 3, 397$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4764=2^{2}\cdot 3\cdot 397\)
Dirichlet character group:    $\lbrace$$\chi_{4764}(1,·)$, $\chi_{4764}(3239,·)$, $\chi_{4764}(4127,·)$, $\chi_{4764}(1225,·)$, $\chi_{4764}(971,·)$, $\chi_{4764}(793,·)$, $\chi_{4764}(4333,·)$, $\chi_{4764}(2999,·)$, $\chi_{4764}(4729,·)$, $\chi_{4764}(731,·)$, $\chi_{4764}(829,·)$, $\chi_{4764}(4607,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{18} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{432} a^{6} - \frac{7}{144} a^{4} - \frac{7}{48} a^{2} - \frac{1}{4}$, $\frac{1}{864} a^{7} - \frac{7}{288} a^{5} - \frac{7}{96} a^{3} + \frac{3}{8} a$, $\frac{1}{124416} a^{8} + \frac{5}{41472} a^{6} - \frac{35}{1536} a^{4} + \frac{17}{192} a^{2} + \frac{1}{96}$, $\frac{1}{248832} a^{9} + \frac{5}{82944} a^{7} - \frac{35}{3072} a^{5} - \frac{47}{384} a^{3} + \frac{1}{192} a$, $\frac{1}{7844570457071616} a^{10} - \frac{12392807}{10501433008128} a^{8} + \frac{55372841153}{871618939674624} a^{6} + \frac{1072585132343}{24211637213184} a^{4} - \frac{239058468461}{6052909303296} a^{2} - \frac{77602350221}{504409108608}$, $\frac{1}{15689140914143232} a^{11} - \frac{12392807}{21002866016256} a^{9} + \frac{55372841153}{1743237879349248} a^{7} + \frac{1072585132343}{48423274426368} a^{5} - \frac{239058468461}{12105818606592} a^{3} - \frac{77602350221}{1008818217216} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{11908}$, which has order $12193792$ (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:  \( 406436.74206624634 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

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

Intermediate fields

\(\Q(\sqrt{397}) \), 3.3.157609.1, 4.0.9010191312.1, 6.6.9861716961757.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
397Data not computed