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Properties

Label 12.0.59469951837...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{10}\cdot 47^{6}$
Root discriminant $116.02$
Ramified primes $5, 7, 47$
Class number $186628$ (GRH)
Class group $[2, 93314]$ (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![19664734321, -2683767652, 2829532400, -185075146, 160998290, -4781629, 4441807, -62203, 61794, -409, 408, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 408*x^10 - 409*x^9 + 61794*x^8 - 62203*x^7 + 4441807*x^6 - 4781629*x^5 + 160998290*x^4 - 185075146*x^3 + 2829532400*x^2 - 2683767652*x + 19664734321)
 
gp: K = bnfinit(x^12 - x^11 + 408*x^10 - 409*x^9 + 61794*x^8 - 62203*x^7 + 4441807*x^6 - 4781629*x^5 + 160998290*x^4 - 185075146*x^3 + 2829532400*x^2 - 2683767652*x + 19664734321, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 408 x^{10} - 409 x^{9} + 61794 x^{8} - 62203 x^{7} + 4441807 x^{6} - 4781629 x^{5} + 160998290 x^{4} - 185075146 x^{3} + 2829532400 x^{2} - 2683767652 x + 19664734321 \)

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:  \(5946995183757601408203125=5^{9}\cdot 7^{10}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.02$
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:  $5, 7, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1645=5\cdot 7\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{1645}(1,·)$, $\chi_{1645}(1411,·)$, $\chi_{1645}(657,·)$, $\chi_{1645}(328,·)$, $\chi_{1645}(1129,·)$, $\chi_{1645}(424,·)$, $\chi_{1645}(659,·)$, $\chi_{1645}(563,·)$, $\chi_{1645}(471,·)$, $\chi_{1645}(187,·)$, $\chi_{1645}(892,·)$, $\chi_{1645}(1503,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{655906439} a^{10} + \frac{79277244}{655906439} a^{9} - \frac{306382675}{655906439} a^{8} + \frac{137289585}{655906439} a^{7} - \frac{314392683}{655906439} a^{6} - \frac{70168565}{655906439} a^{5} + \frac{123136476}{655906439} a^{4} + \frac{109001415}{655906439} a^{3} - \frac{188909951}{655906439} a^{2} - \frac{37119783}{655906439} a - \frac{235917496}{655906439}$, $\frac{1}{95602051886011396697014938633903726829} a^{11} + \frac{46662086807887325085150436027}{95602051886011396697014938633903726829} a^{10} - \frac{517714837474531354083488088801245869}{95602051886011396697014938633903726829} a^{9} + \frac{11129672868465445749202495832977685787}{95602051886011396697014938633903726829} a^{8} + \frac{46326298376612809401159884983462722497}{95602051886011396697014938633903726829} a^{7} + \frac{19051895660193240698179151601734694973}{95602051886011396697014938633903726829} a^{6} + \frac{31347884529124285522917752564564854610}{95602051886011396697014938633903726829} a^{5} - \frac{22014287106930711695587889380971616257}{95602051886011396697014938633903726829} a^{4} + \frac{848246972833103150971365059521608516}{95602051886011396697014938633903726829} a^{3} - \frac{1492831661335950352953301053913927799}{95602051886011396697014938633903726829} a^{2} + \frac{748082961534589547092984143767588787}{95602051886011396697014938633903726829} a + \frac{1870152762751794710289541216}{4861598958085989424083353149}$

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

Class group and class number

$C_{2}\times C_{93314}$, which has order $186628$ (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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.13530125.1, 6.6.300125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$47$47.12.6.2$x^{12} - 229345007 x^{2} + 53896076645$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$