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Properties

Label 12.0.31701123988...1984.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 3^{6}\cdot 7^{10}\cdot 13^{11}$
Root discriminant $619.09$
Ramified primes $2, 3, 7, 13$
Class number $31804608$ (GRH)
Class group $[2, 2, 2, 814, 4884]$ (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![449513064, 0, 23518325376, 0, 2733815448, 0, 54427464, 0, 384930, 0, 1092, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 1092*x^10 + 384930*x^8 + 54427464*x^6 + 2733815448*x^4 + 23518325376*x^2 + 449513064)
 
gp: K = bnfinit(x^12 + 1092*x^10 + 384930*x^8 + 54427464*x^6 + 2733815448*x^4 + 23518325376*x^2 + 449513064, 1)
 

Normalized defining polynomial

\( x^{12} + 1092 x^{10} + 384930 x^{8} + 54427464 x^{6} + 2733815448 x^{4} + 23518325376 x^{2} + 449513064 \)

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:  \(3170112398857312997665091547561984=2^{33}\cdot 3^{6}\cdot 7^{10}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $619.09$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4368=2^{4}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4368}(1,·)$, $\chi_{4368}(2021,·)$, $\chi_{4368}(3649,·)$, $\chi_{4368}(361,·)$, $\chi_{4368}(4301,·)$, $\chi_{4368}(125,·)$, $\chi_{4368}(1537,·)$, $\chi_{4368}(629,·)$, $\chi_{4368}(2521,·)$, $\chi_{4368}(121,·)$, $\chi_{4368}(1853,·)$, $\chi_{4368}(1445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18} a^{4}$, $\frac{1}{18} a^{5}$, $\frac{1}{378} a^{6}$, $\frac{1}{4158} a^{7} - \frac{1}{198} a^{5} - \frac{1}{11} a^{3} - \frac{5}{11} a$, $\frac{1}{1122660} a^{8} + \frac{4}{31185} a^{6} + \frac{1}{55} a^{4} + \frac{23}{495} a^{2} - \frac{22}{45}$, $\frac{1}{1122660} a^{9} - \frac{1}{8910} a^{7} + \frac{23}{990} a^{5} + \frac{68}{495} a^{3} - \frac{17}{495} a$, $\frac{1}{1141121284865100} a^{10} - \frac{8199377}{54339108803100} a^{8} - \frac{2865239173}{21131875645650} a^{6} - \frac{6508951081}{503139896325} a^{4} + \frac{11688733643}{100627979265} a^{2} + \frac{6075646634}{15246663525}$, $\frac{1}{1141121284865100} a^{11} - \frac{8199377}{54339108803100} a^{9} + \frac{1108491001}{10565937822825} a^{7} - \frac{18100123337}{1006279792650} a^{5} + \frac{2540735528}{100627979265} a^{3} - \frac{9401204651}{167713298775} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{814}\times C_{4884}$, which has order $31804608$ (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:  \( 67617.0856763039 \) (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

\(\Q(\sqrt{26}) \), 3.3.8281.2, 4.0.1984260096.2, 6.6.456434940416.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.33.340$x^{12} - 8 x^{10} - 28 x^{8} - 8 x^{6} + 20 x^{4} + 16 x^{2} - 24$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$7$7.12.10.6$x^{12} - 217 x^{6} + 11907$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$13$13.12.11.8$x^{12} + 104$$12$$1$$11$$C_{12}$$[\ ]_{12}$