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Properties

Label 12.0.11848149020...6176.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{18}\cdot 67^{10}$
Root discriminant $690.99$
Ramified primes $2, 3, 67$
Class number $14086800$ (GRH)
Class group $[30, 469560]$ (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![348339581209, 0, 48604503852, 0, 2115867705, 0, 33685456, 0, 242406, 0, 804, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 804*x^10 + 242406*x^8 + 33685456*x^6 + 2115867705*x^4 + 48604503852*x^2 + 348339581209)
 
gp: K = bnfinit(x^12 + 804*x^10 + 242406*x^8 + 33685456*x^6 + 2115867705*x^4 + 48604503852*x^2 + 348339581209, 1)
 

Normalized defining polynomial

\( x^{12} + 804 x^{10} + 242406 x^{8} + 33685456 x^{6} + 2115867705 x^{4} + 48604503852 x^{2} + 348339581209 \)

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:  \(11848149020404956572813211180466176=2^{24}\cdot 3^{18}\cdot 67^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $690.99$
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:  $2, 3, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4824=2^{3}\cdot 3^{2}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{4824}(2977,·)$, $\chi_{4824}(4259,·)$, $\chi_{4824}(3589,·)$, $\chi_{4824}(1,·)$, $\chi_{4824}(841,·)$, $\chi_{4824}(2411,·)$, $\chi_{4824}(3887,·)$, $\chi_{4824}(1571,·)$, $\chi_{4824}(3349,·)$, $\chi_{4824}(3119,·)$, $\chi_{4824}(4117,·)$, $\chi_{4824}(3647,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{67067} a^{6} + \frac{6}{1001} a^{4} - \frac{398}{1001} a^{2} - \frac{31}{1001}$, $\frac{1}{590793203} a^{7} - \frac{88082}{8817809} a^{5} - \frac{3055450}{8817809} a^{3} - \frac{1936966}{8817809} a$, $\frac{1}{590793203} a^{8} + \frac{8}{8817809} a^{6} - \frac{416358}{1259687} a^{4} + \frac{3410097}{8817809} a^{2} + \frac{251}{1001}$, $\frac{1}{590793203} a^{9} + \frac{208401}{8817809} a^{5} + \frac{78371}{678293} a^{3} - \frac{76627}{8817809} a$, $\frac{1}{590793203} a^{10} + \frac{86}{84399029} a^{6} - \frac{3394486}{8817809} a^{4} + \frac{240739}{1259687} a^{2} + \frac{86}{1001}$, $\frac{1}{590793203} a^{11} - \frac{297816}{801619} a^{5} - \frac{1048}{4979} a^{3} + \frac{2860318}{8817809} a$

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

Class group and class number

$C_{30}\times C_{469560}$, which has order $14086800$ (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:  \( 1581818.8105486752 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-402}) \), \(\Q(\sqrt{-134}) \), 3.3.363609.1, \(\Q(\sqrt{3}, \sqrt{-134})\), 6.6.25384608937152.1, 6.0.13606150390313472.2, 6.0.4535383463437824.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
$67$67.12.10.2$x^{12} + 1541 x^{6} + 646416$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$