↑

Properties

Label 12.0.15201924401...000.15
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 19^{10}$
Root discriminant $270.29$
Ramified primes $2, 3, 5, 19$
Class number $1707264$ (GRH)
Class group $[2, 6, 12, 11856]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2100205584, 0, 588796776, 0, 57419577, 0, 1542876, 0, 26334, 0, 228, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 228*x^10 + 26334*x^8 + 1542876*x^6 + 57419577*x^4 + 588796776*x^2 + 2100205584)
 
gp: K = bnfinit(x^12 + 228*x^10 + 26334*x^8 + 1542876*x^6 + 57419577*x^4 + 588796776*x^2 + 2100205584, 1)
 

Normalized defining polynomial

\( x^{12} + 228 x^{10} + 26334 x^{8} + 1542876 x^{6} + 57419577 x^{4} + 588796776 x^{2} + 2100205584 \)

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:  \(152019244012074463020096000000=2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.29$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3420=2^{2}\cdot 3^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3420}(1,·)$, $\chi_{3420}(3419,·)$, $\chi_{3420}(1319,·)$, $\chi_{3420}(2089,·)$, $\chi_{3420}(1451,·)$, $\chi_{3420}(1331,·)$, $\chi_{3420}(2401,·)$, $\chi_{3420}(1969,·)$, $\chi_{3420}(2291,·)$, $\chi_{3420}(2101,·)$, $\chi_{3420}(1129,·)$, $\chi_{3420}(1019,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{3648} a^{6} - \frac{1}{32} a^{4} + \frac{25}{64} a^{2} + \frac{5}{16}$, $\frac{1}{7296} a^{7} + \frac{3}{64} a^{5} - \frac{31}{128} a^{3} + \frac{13}{32} a$, $\frac{1}{284544} a^{8} + \frac{1}{23712} a^{6} + \frac{3}{128} a^{4} + \frac{1213}{2496} a^{2} - \frac{35}{208}$, $\frac{1}{19064448} a^{9} + \frac{41}{3177408} a^{7} - \frac{217}{8576} a^{5} - \frac{14935}{83616} a^{3} + \frac{449}{1742} a$, $\frac{1}{316622352384} a^{10} - \frac{34691}{105540784128} a^{8} - \frac{528183}{11726753792} a^{6} + \frac{327142733}{5554778112} a^{4} - \frac{19941777}{77149696} a^{2} - \frac{43145}{575744}$, $\frac{1}{18047474085888} a^{11} - \frac{5905}{316622352384} a^{9} + \frac{2060147}{35180261376} a^{7} + \frac{1426957709}{316622352384} a^{5} - \frac{133141193}{694347264} a^{3} + \frac{44420087}{115724544} a$

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

Class group and class number

$C_{2}\times C_{6}\times C_{12}\times C_{11856}$, which has order $1707264$ (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:  \( 475218.31627778086 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-95}) \), 3.3.29241.1, \(\Q(\sqrt{3}, \sqrt{-95})\), 6.0.389896452936000.2, 6.6.164166927552.2, 6.0.2030710692375.4

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.12.10.1$x^{12} - 171 x^{6} + 23104$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$