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Properties

Label 12.0.27482671235...856.18
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 19^{10}$
Root discriminant $611.77$
Ramified primes $2, 3, 7, 19$
Class number $12607488$ (GRH)
Class group $[2, 2, 2, 2, 6, 12, 12, 912]$ (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![624972464704, 0, 50119658820, 0, 933713865, 0, -1550780, 0, 64638, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 64638*x^8 - 1550780*x^6 + 933713865*x^4 + 50119658820*x^2 + 624972464704)
 
gp: K = bnfinit(x^12 + 64638*x^8 - 1550780*x^6 + 933713865*x^4 + 50119658820*x^2 + 624972464704, 1)
 

Normalized defining polynomial

\( x^{12} + 64638 x^{8} - 1550780 x^{6} + 933713865 x^{4} + 50119658820 x^{2} + 624972464704 \)

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:  \(2748267123526559548681146214649856=2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $611.77$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(4225,·)$, $\chi_{4788}(2083,·)$, $\chi_{4788}(1,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(331,·)$, $\chi_{4788}(2957,·)$, $\chi_{4788}(2063,·)$, $\chi_{4788}(311,·)$, $\chi_{4788}(1433,·)$, $\chi_{4788}(379,·)$, $\chi_{4788}(2393,·)$, $\chi_{4788}(2015,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{1064} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{2128} a^{7} - \frac{1}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{289408} a^{8} + \frac{59}{144704} a^{6} - \frac{139}{2176} a^{4} - \frac{235}{544} a^{2} - \frac{11}{34}$, $\frac{1}{215030144} a^{9} + \frac{11687}{107515072} a^{7} - \frac{84459}{1616768} a^{5} - \frac{83433}{404192} a^{3} + \frac{2307}{50524} a$, $\frac{1}{39065905893883904} a^{10} - \frac{66955211357}{39065905893883904} a^{8} + \frac{434356335831}{2297994464346112} a^{6} - \frac{34770019774363}{293728615743488} a^{4} - \frac{6052616929231}{73432153935872} a^{2} - \frac{3008808737}{6176998144}$, $\frac{1}{5195765483886559232} a^{11} - \frac{67672441}{39065905893883904} a^{9} + \frac{12981776957}{328284923478016} a^{7} + \frac{651271193708773}{39065905893883904} a^{5} - \frac{5517160833731}{73432153935872} a^{3} - \frac{1161908228315}{4589509620992} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{12}\times C_{12}\times C_{912}$, which has order $12607488$ (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:  \( 6344072.157047723 \) (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

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{-399}) \), 3.3.1432809.2, \(\Q(\sqrt{19}, \sqrt{-21})\), 6.0.2759153551366464.5, 6.6.2496377022664896.4, 6.0.819123710561919.3

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.6.5.1$x^{6} - 304$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.1$x^{6} - 304$$6$$1$$5$$C_6$$[\ ]_{6}$