↑

Properties

Label 12.0.97816263424...5561.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 41^{6}$
Root discriminant $56.13$
Ramified primes $3, 7, 41$
Class number $1568$ (GRH)
Class group $[2, 2, 2, 14, 14]$ (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![3282511, 1806919, 1416470, 278581, 577579, -8490, 76481, -1920, 4369, -79, 110, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 110*x^10 - 79*x^9 + 4369*x^8 - 1920*x^7 + 76481*x^6 - 8490*x^5 + 577579*x^4 + 278581*x^3 + 1416470*x^2 + 1806919*x + 3282511)
 
gp: K = bnfinit(x^12 - x^11 + 110*x^10 - 79*x^9 + 4369*x^8 - 1920*x^7 + 76481*x^6 - 8490*x^5 + 577579*x^4 + 278581*x^3 + 1416470*x^2 + 1806919*x + 3282511, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 110 x^{10} - 79 x^{9} + 4369 x^{8} - 1920 x^{7} + 76481 x^{6} - 8490 x^{5} + 577579 x^{4} + 278581 x^{3} + 1416470 x^{2} + 1806919 x + 3282511 \)

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:  \(978162634246022205561=3^{6}\cdot 7^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.13$
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:  $3, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(861=3\cdot 7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{861}(1,·)$, $\chi_{861}(739,·)$, $\chi_{861}(737,·)$, $\chi_{861}(40,·)$, $\chi_{861}(491,·)$, $\chi_{861}(206,·)$, $\chi_{861}(368,·)$, $\chi_{861}(83,·)$, $\chi_{861}(247,·)$, $\chi_{861}(409,·)$, $\chi_{861}(698,·)$, $\chi_{861}(286,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1651} a^{7} + \frac{70}{1651} a^{5} - \frac{251}{1651} a^{3} + \frac{396}{1651} a + \frac{108}{1651}$, $\frac{1}{1651} a^{8} + \frac{70}{1651} a^{6} - \frac{251}{1651} a^{4} + \frac{396}{1651} a^{2} + \frac{108}{1651} a$, $\frac{1}{18161} a^{9} + \frac{2}{18161} a^{7} - \frac{1}{11} a^{6} - \frac{5011}{18161} a^{5} - \frac{5}{11} a^{4} - \frac{5650}{18161} a^{3} + \frac{6712}{18161} a^{2} + \frac{1139}{18161} a + \frac{5864}{18161}$, $\frac{1}{134982195491} a^{10} + \frac{306694}{12271108681} a^{9} + \frac{24052514}{134982195491} a^{8} - \frac{28153687}{134982195491} a^{7} - \frac{46702536183}{134982195491} a^{6} + \frac{13209257606}{134982195491} a^{5} + \frac{9374292921}{134982195491} a^{4} + \frac{34266138975}{134982195491} a^{3} + \frac{45800564475}{134982195491} a^{2} - \frac{33663494314}{134982195491} a + \frac{5279729930}{12271108681}$, $\frac{1}{134982195491} a^{11} + \frac{379141}{134982195491} a^{9} + \frac{4749414}{134982195491} a^{8} - \frac{146511}{134982195491} a^{7} - \frac{18920911962}{134982195491} a^{6} + \frac{26815173927}{134982195491} a^{5} + \frac{28760519153}{134982195491} a^{4} - \frac{45395049686}{134982195491} a^{3} + \frac{247528643}{1062851933} a^{2} + \frac{2280920555}{134982195491} a - \frac{29746024064}{134982195491}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{14}$, which has order $1568$ (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:  \( 140.7987960054707 \) (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

\(\Q(\sqrt{-123}) \), \(\Q(\sqrt{-287}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-123})\), 6.0.4467941667.1, 6.0.1158355247.1, \(\Q(\zeta_{21})^+\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$