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Properties

Label 12.0.30592461885...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 31^{10}$
Root discriminant $286.51$
Ramified primes $2, 3, 5, 31$
Class number $1889920$ (GRH)
Class group $[2, 2, 2, 2, 2, 59060]$ (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![1434765312000, 0, 119563776000, 0, 3487276800, 0, 46872000, 0, 306900, 0, 930, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 930*x^10 + 306900*x^8 + 46872000*x^6 + 3487276800*x^4 + 119563776000*x^2 + 1434765312000)
 
gp: K = bnfinit(x^12 + 930*x^10 + 306900*x^8 + 46872000*x^6 + 3487276800*x^4 + 119563776000*x^2 + 1434765312000, 1)
 

Normalized defining polynomial

\( x^{12} + 930 x^{10} + 306900 x^{8} + 46872000 x^{6} + 3487276800 x^{4} + 119563776000 x^{2} + 1434765312000 \)

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:  \(305924618859010011648000000000=2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $286.51$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(3653,·)$, $\chi_{3720}(769,·)$, $\chi_{3720}(3533,·)$, $\chi_{3720}(557,·)$, $\chi_{3720}(3601,·)$, $\chi_{3720}(1489,·)$, $\chi_{3720}(533,·)$, $\chi_{3720}(1369,·)$, $\chi_{3720}(3001,·)$, $\chi_{3720}(1277,·)$, $\chi_{3720}(677,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{180} a^{4}$, $\frac{1}{360} a^{5} - \frac{1}{12} a^{3} - \frac{1}{2} a$, $\frac{1}{133920} a^{6} + \frac{1}{720} a^{4} - \frac{1}{24} a^{2}$, $\frac{1}{267840} a^{7} + \frac{1}{1440} a^{5} + \frac{1}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{128563200} a^{8} + \frac{1}{1428480} a^{6} - \frac{7}{7680} a^{4} + \frac{5}{192} a^{2} - \frac{1}{8}$, $\frac{1}{257126400} a^{9} + \frac{1}{2856960} a^{7} - \frac{7}{15360} a^{5} + \frac{5}{384} a^{3} + \frac{7}{16} a$, $\frac{1}{3085516800} a^{10} - \frac{1}{514252800} a^{8} - \frac{59}{17141760} a^{6} + \frac{13}{23040} a^{4} - \frac{5}{192} a^{2} - \frac{1}{2}$, $\frac{1}{6171033600} a^{11} - \frac{1}{1028505600} a^{9} - \frac{59}{34283520} a^{7} + \frac{13}{46080} a^{5} - \frac{5}{384} a^{3} - \frac{1}{4} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{59060}$, which has order $1889920$ (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:  \( 4882.160216514567 \) (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

\(\Q(\sqrt{5}) \), 3.3.961.1, 4.0.69192000.5, 6.6.115440125.1

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 R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$3$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$31$31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$