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Properties

Label 12.0.24955689687...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 53^{6}$
Root discriminant $89.08$
Ramified primes $5, 7, 53$
Class number $19330$ (GRH)
Class group $[19330]$ (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![995262101, 45546027, 183583288, 6384683, 14417911, 294994, 612804, 3228, 14713, -56, 188, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 188*x^10 - 56*x^9 + 14713*x^8 + 3228*x^7 + 612804*x^6 + 294994*x^5 + 14417911*x^4 + 6384683*x^3 + 183583288*x^2 + 45546027*x + 995262101)
 
gp: K = bnfinit(x^12 - x^11 + 188*x^10 - 56*x^9 + 14713*x^8 + 3228*x^7 + 612804*x^6 + 294994*x^5 + 14417911*x^4 + 6384683*x^3 + 183583288*x^2 + 45546027*x + 995262101, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 188 x^{10} - 56 x^{9} + 14713 x^{8} + 3228 x^{7} + 612804 x^{6} + 294994 x^{5} + 14417911 x^{4} + 6384683 x^{3} + 183583288 x^{2} + 45546027 x + 995262101 \)

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:  \(249556896876602205078125=5^{9}\cdot 7^{8}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.08$
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:  $5, 7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1855=5\cdot 7\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{1855}(1,·)$, $\chi_{1855}(1061,·)$, $\chi_{1855}(582,·)$, $\chi_{1855}(1642,·)$, $\chi_{1855}(688,·)$, $\chi_{1855}(849,·)$, $\chi_{1855}(1591,·)$, $\chi_{1855}(953,·)$, $\chi_{1855}(1114,·)$, $\chi_{1855}(317,·)$, $\chi_{1855}(158,·)$, $\chi_{1855}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{209} a^{9} + \frac{39}{209} a^{8} - \frac{37}{209} a^{7} - \frac{51}{209} a^{6} + \frac{2}{19} a^{5} + \frac{31}{209} a^{4} + \frac{74}{209} a^{3} + \frac{14}{209} a^{2} - \frac{37}{209} a + \frac{49}{209}$, $\frac{1}{209} a^{10} - \frac{5}{11} a^{8} - \frac{71}{209} a^{7} - \frac{79}{209} a^{6} + \frac{9}{209} a^{5} - \frac{90}{209} a^{4} + \frac{54}{209} a^{3} + \frac{4}{19} a^{2} + \frac{29}{209} a - \frac{30}{209}$, $\frac{1}{10079519574894325801696313989400219} a^{11} + \frac{18582277396180423770678921241332}{10079519574894325801696313989400219} a^{10} - \frac{5925115401404199052584746049789}{10079519574894325801696313989400219} a^{9} + \frac{2552908058644790836751791586504978}{10079519574894325801696313989400219} a^{8} - \frac{3157497589831483363364156933448099}{10079519574894325801696313989400219} a^{7} - \frac{2116826956284473994569375970127843}{10079519574894325801696313989400219} a^{6} + \frac{2717750178985584033459703184849325}{10079519574894325801696313989400219} a^{5} - \frac{3304666033383312715546128682901922}{10079519574894325801696313989400219} a^{4} - \frac{799804050023523644197461614625857}{10079519574894325801696313989400219} a^{3} - \frac{2635549027729358191424336229855138}{10079519574894325801696313989400219} a^{2} - \frac{3048398830800284506688747315641623}{10079519574894325801696313989400219} a + \frac{2473245557841496023754671602886876}{10079519574894325801696313989400219}$

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

Class group and class number

$C_{19330}$, which has order $19330$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.351125.1, 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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$53$53.12.6.2$x^{12} - 418195493 x^{2} + 177314889032$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$