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Properties

Label 12.0.20550188590...056.16
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{6}\cdot 7^{10}\cdot 29^{6}$
Root discriminant $188.83$
Ramified primes $2, 3, 7, 29$
Class number $1557504$ (GRH)
Class group $[2, 2, 2, 4, 52, 936]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![371530478089, 0, 30891844344, 0, 1039094758, 0, 16983632, 0, 143581, 0, 604, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 604*x^10 + 143581*x^8 + 16983632*x^6 + 1039094758*x^4 + 30891844344*x^2 + 371530478089)
 
gp: K = bnfinit(x^12 + 604*x^10 + 143581*x^8 + 16983632*x^6 + 1039094758*x^4 + 30891844344*x^2 + 371530478089, 1)
 

Normalized defining polynomial

\( x^{12} + 604 x^{10} + 143581 x^{8} + 16983632 x^{6} + 1039094758 x^{4} + 30891844344 x^{2} + 371530478089 \)

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:  \(2055018859092572865671725056=2^{24}\cdot 3^{6}\cdot 7^{10}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.83$
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:  $2, 3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4872=2^{3}\cdot 3\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4872}(1,·)$, $\chi_{4872}(2435,·)$, $\chi_{4872}(2437,·)$, $\chi_{4872}(4871,·)$, $\chi_{4872}(1739,·)$, $\chi_{4872}(4175,·)$, $\chi_{4872}(3827,·)$, $\chi_{4872}(1045,·)$, $\chi_{4872}(697,·)$, $\chi_{4872}(3481,·)$, $\chi_{4872}(1391,·)$, $\chi_{4872}(3133,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{609533} a^{7} + \frac{301}{609533} a^{5} + \frac{25886}{609533} a^{3} - \frac{52984}{609533} a$, $\frac{1}{62515533079} a^{8} + \frac{1251371550}{62515533079} a^{6} + \frac{10276142733}{62515533079} a^{4} + \frac{6396995851}{62515533079} a^{2} - \frac{2287}{102563}$, $\frac{1}{62515533079} a^{9} + \frac{387}{62515533079} a^{7} + \frac{8706621144}{62515533079} a^{5} - \frac{3550794645}{62515533079} a^{3} - \frac{27724898398}{62515533079} a$, $\frac{1}{62515533079} a^{10} + \frac{24550095926}{62515533079} a^{6} + \frac{20576084740}{62515533079} a^{4} - \frac{2740969575}{62515533079} a^{2} - \frac{37998}{102563}$, $\frac{1}{62515533079} a^{11} + \frac{868}{62515533079} a^{7} + \frac{7830375604}{62515533079} a^{5} + \frac{26407640151}{62515533079} a^{3} - \frac{21621256615}{62515533079} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{52}\times C_{936}$, which has order $1557504$ (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:  \( 279.1500271937239 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-1218}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-609}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-609})\), 6.0.5666539479552.2, 6.6.1229312.1, 6.0.708317434944.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$