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Properties

Label 12.0.21876410091...000.36
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 13^{10}$
Root discriminant $278.61$
Ramified primes $2, 3, 5, 13$
Class number $1303200$ (GRH)
Class group $[2, 30, 21720]$ (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![5314031919, 458844750, 462875499, 13489866, 17475255, 83538, 471412, 3198, 8703, -52, 102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 102*x^10 - 52*x^9 + 8703*x^8 + 3198*x^7 + 471412*x^6 + 83538*x^5 + 17475255*x^4 + 13489866*x^3 + 462875499*x^2 + 458844750*x + 5314031919)
 
gp: K = bnfinit(x^12 + 102*x^10 - 52*x^9 + 8703*x^8 + 3198*x^7 + 471412*x^6 + 83538*x^5 + 17475255*x^4 + 13489866*x^3 + 462875499*x^2 + 458844750*x + 5314031919, 1)
 

Normalized defining polynomial

\( x^{12} + 102 x^{10} - 52 x^{9} + 8703 x^{8} + 3198 x^{7} + 471412 x^{6} + 83538 x^{5} + 17475255 x^{4} + 13489866 x^{3} + 462875499 x^{2} + 458844750 x + 5314031919 \)

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:  \(218764100914962817683456000000=2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.61$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1921,·)$, $\chi_{4680}(1829,·)$, $\chi_{4680}(1,·)$, $\chi_{4680}(4469,·)$, $\chi_{4680}(3721,·)$, $\chi_{4680}(2401,·)$, $\chi_{4680}(1681,·)$, $\chi_{4680}(1589,·)$, $\chi_{4680}(2521,·)$, $\chi_{4680}(989,·)$, $\chi_{4680}(1109,·)$, $\chi_{4680}(3509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{69} a^{8} - \frac{10}{69} a^{7} + \frac{1}{69} a^{6} + \frac{10}{69} a^{5} + \frac{5}{69} a^{4} + \frac{4}{69} a^{3} + \frac{11}{23} a^{2} + \frac{4}{23} a - \frac{11}{23}$, $\frac{1}{207} a^{9} - \frac{1}{207} a^{8} - \frac{20}{207} a^{7} + \frac{19}{207} a^{6} + \frac{26}{207} a^{5} - \frac{89}{207} a^{4} + \frac{1}{3} a^{3} + \frac{34}{69} a^{2} - \frac{7}{23} a - \frac{10}{23}$, $\frac{1}{1433061} a^{10} + \frac{880}{477687} a^{9} + \frac{265}{159229} a^{8} + \frac{185159}{1433061} a^{7} - \frac{8489}{159229} a^{6} + \frac{5041}{477687} a^{5} - \frac{35987}{204723} a^{4} - \frac{959}{2967} a^{3} + \frac{31321}{68241} a^{2} + \frac{1847}{159229} a + \frac{9454}{22747}$, $\frac{1}{53871028522932843998846171577} a^{11} + \frac{252733845958113662426}{53871028522932843998846171577} a^{10} - \frac{44821370373961878313588642}{53871028522932843998846171577} a^{9} - \frac{111044286916753626573124378}{17957009507644281332948723859} a^{8} - \frac{350108121410969537228931740}{5985669835881427110982907953} a^{7} + \frac{8974683878128728669381860417}{53871028522932843998846171577} a^{6} - \frac{8618484834916796081235799741}{53871028522932843998846171577} a^{5} - \frac{1540332616856404851215797439}{7695861217561834856978024511} a^{4} + \frac{188287552211777354718184836}{855095690840203872997558279} a^{3} + \frac{1940453979822139800784132952}{17957009507644281332948723859} a^{2} - \frac{1466605362172215752736328236}{5985669835881427110982907953} a + \frac{548031844592178179810362}{3327220586926863319056647}$

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

Class group and class number

$C_{2}\times C_{30}\times C_{21720}$, which has order $1303200$ (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:  \( 9116.238746847283 \) (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

\(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-390}) \), \(\Q(\sqrt{13}) \), 3.3.13689.2, \(\Q(\sqrt{13}, \sqrt{-30})\), 6.0.35978634432000.4, 6.0.467722247616000.1, 6.6.2436053373.2

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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$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}$
$13$13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$