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Properties

Label 12.0.24014839788...4816.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 73^{6}$
Root discriminant $88.79$
Ramified primes $2, 3, 73$
Class number $11648$ (GRH)
Class group $[2, 2, 2, 2, 2, 364]$ (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![78446449, 0, 19219578, 0, 1988841, 0, 115271, 0, 4164, 0, 93, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 93*x^10 + 4164*x^8 + 115271*x^6 + 1988841*x^4 + 19219578*x^2 + 78446449)
 
gp: K = bnfinit(x^12 + 93*x^10 + 4164*x^8 + 115271*x^6 + 1988841*x^4 + 19219578*x^2 + 78446449, 1)
 

Normalized defining polynomial

\( x^{12} + 93 x^{10} + 4164 x^{8} + 115271 x^{6} + 1988841 x^{4} + 19219578 x^{2} + 78446449 \)

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:  \(240148397880610960674816=2^{12}\cdot 3^{18}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.79$
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:  $2, 3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2628=2^{2}\cdot 3^{2}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{2628}(1,·)$, $\chi_{2628}(1459,·)$, $\chi_{2628}(1313,·)$, $\chi_{2628}(1607,·)$, $\chi_{2628}(583,·)$, $\chi_{2628}(2189,·)$, $\chi_{2628}(877,·)$, $\chi_{2628}(2483,·)$, $\chi_{2628}(437,·)$, $\chi_{2628}(1753,·)$, $\chi_{2628}(731,·)$, $\chi_{2628}(2335,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{37} a^{8} - \frac{1}{37} a^{6} + \frac{13}{37} a^{4} + \frac{5}{37} a^{2} + \frac{10}{37}$, $\frac{1}{37} a^{9} - \frac{1}{37} a^{7} + \frac{13}{37} a^{5} + \frac{5}{37} a^{3} + \frac{10}{37} a$, $\frac{1}{6273924499} a^{10} - \frac{52762380}{6273924499} a^{8} - \frac{2132892469}{6273924499} a^{6} + \frac{2597138409}{6273924499} a^{4} + \frac{1642165230}{6273924499} a^{2} - \frac{473107472}{6273924499}$, $\frac{1}{55568149287643} a^{11} + \frac{96599588010}{55568149287643} a^{9} + \frac{21652910995174}{55568149287643} a^{7} + \frac{26549267348948}{55568149287643} a^{5} + \frac{8986071613325}{55568149287643} a^{3} - \frac{13840411421212}{55568149287643} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{364}$, which has order $11648$ (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:  \( 325.67540279491664 \) (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

\(\Q(\sqrt{-73}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-219}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-73})\), 6.0.163349794368.4, \(\Q(\zeta_{36})^+\), 6.0.7657021611.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$73$73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$