↑

Properties

Label 12.0.14082882354...3117.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 13^{11}\cdot 47^{6}$
Root discriminant $124.66$
Ramified primes $3, 13, 47$
Class number $138560$ (GRH)
Class group $[2, 2, 2, 2, 8660]$ (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28956826081, -5059372956, 5059372956, -279882331, 279882331, -6768581, 6768581, -80081, 80081, -456, 456, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 456*x^10 - 456*x^9 + 80081*x^8 - 80081*x^7 + 6768581*x^6 - 6768581*x^5 + 279882331*x^4 - 279882331*x^3 + 5059372956*x^2 - 5059372956*x + 28956826081)
 
gp: K = bnfinit(x^12 - x^11 + 456*x^10 - 456*x^9 + 80081*x^8 - 80081*x^7 + 6768581*x^6 - 6768581*x^5 + 279882331*x^4 - 279882331*x^3 + 5059372956*x^2 - 5059372956*x + 28956826081, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 456 x^{10} - 456 x^{9} + 80081 x^{8} - 80081 x^{7} + 6768581 x^{6} - 6768581 x^{5} + 279882331 x^{4} - 279882331 x^{3} + 5059372956 x^{2} - 5059372956 x + 28956826081 \)

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:  \(14082882354952696422423117=3^{6}\cdot 13^{11}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.66$
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:  $3, 13, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1833=3\cdot 13\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{1833}(704,·)$, $\chi_{1833}(1,·)$, $\chi_{1833}(706,·)$, $\chi_{1833}(422,·)$, $\chi_{1833}(1409,·)$, $\chi_{1833}(142,·)$, $\chi_{1833}(1268,·)$, $\chi_{1833}(1270,·)$, $\chi_{1833}(281,·)$, $\chi_{1833}(986,·)$, $\chi_{1833}(283,·)$, $\chi_{1833}(1693,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3049925761} a^{7} - \frac{1207661917}{3049925761} a^{6} + \frac{245}{3049925761} a^{5} - \frac{465164407}{3049925761} a^{4} + \frac{17150}{3049925761} a^{3} + \frac{1503237601}{3049925761} a^{2} + \frac{300125}{3049925761} a + \frac{169906244}{3049925761}$, $\frac{1}{3049925761} a^{8} + \frac{280}{3049925761} a^{6} - \frac{430793559}{3049925761} a^{5} + \frac{24500}{3049925761} a^{4} + \frac{859271200}{3049925761} a^{3} + \frac{686000}{3049925761} a^{2} - \frac{424765610}{3049925761} a + \frac{3001250}{3049925761}$, $\frac{1}{3049925761} a^{9} - \frac{827216270}{3049925761} a^{6} - \frac{44100}{3049925761} a^{5} - \frac{41502563}{3049925761} a^{4} - \frac{4116000}{3049925761} a^{3} - \frac{441538872}{3049925761} a^{2} - \frac{81033750}{3049925761} a + \frac{1225063856}{3049925761}$, $\frac{1}{3049925761} a^{10} - \frac{55125}{3049925761} a^{6} + \frac{1331383361}{3049925761} a^{5} - \frac{6431250}{3049925761} a^{4} + \frac{1112777217}{3049925761} a^{3} - \frac{202584375}{3049925761} a^{2} - \frac{548699316}{3049925761} a - \frac{945393750}{3049925761}$, $\frac{1}{3049925761} a^{11} - \frac{302205917}{3049925761} a^{6} + \frac{7074375}{3049925761} a^{5} - \frac{349285931}{3049925761} a^{4} + \frac{742809375}{3049925761} a^{3} - \frac{1058870561}{3049925761} a^{2} + \frac{349368070}{3049925761} a - \frac{240311531}{3049925761}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8660}$, which has order $138560$ (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:  \( 120.784031363 \) (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

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.43678557.2, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$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}$
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$47$47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$