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Properties

Label 12.0.274941996890625.3
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{6}\cdot 17^{6}$
Root discriminant $15.97$
Ramified primes $3, 5, 17$
Class number $4$
Class group $[4]$
Galois group $D_6$ (as 12T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 90, 127, 204, 103, -14, 133, -44, 52, -14, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 13*x^10 - 14*x^9 + 52*x^8 - 44*x^7 + 133*x^6 - 14*x^5 + 103*x^4 + 204*x^3 + 127*x^2 + 90*x + 81)
 
gp: K = bnfinit(x^12 - 3*x^11 + 13*x^10 - 14*x^9 + 52*x^8 - 44*x^7 + 133*x^6 - 14*x^5 + 103*x^4 + 204*x^3 + 127*x^2 + 90*x + 81, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 13 x^{10} - 14 x^{9} + 52 x^{8} - 44 x^{7} + 133 x^{6} - 14 x^{5} + 103 x^{4} + 204 x^{3} + 127 x^{2} + 90 x + 81 \)

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:  \(274941996890625=3^{6}\cdot 5^{6}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.97$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a$, $\frac{1}{9} a^{7} - \frac{1}{9} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{27} a^{9} + \frac{1}{27} a^{7} - \frac{1}{27} a^{6} + \frac{1}{27} a^{5} + \frac{8}{27} a^{4} - \frac{1}{3} a^{3} - \frac{10}{27} a^{2} + \frac{1}{3} a$, $\frac{1}{1053} a^{10} + \frac{14}{1053} a^{9} + \frac{37}{1053} a^{8} - \frac{41}{1053} a^{7} - \frac{22}{1053} a^{6} + \frac{166}{1053} a^{5} + \frac{175}{1053} a^{4} + \frac{197}{1053} a^{3} - \frac{419}{1053} a^{2} + \frac{11}{39} a - \frac{2}{13}$, $\frac{1}{1677429} a^{11} + \frac{400}{1677429} a^{10} + \frac{17843}{1677429} a^{9} - \frac{11711}{559143} a^{8} - \frac{79613}{1677429} a^{7} + \frac{73808}{1677429} a^{6} + \frac{10649}{186381} a^{5} + \frac{659533}{1677429} a^{4} - \frac{102100}{1677429} a^{3} + \frac{502889}{1677429} a^{2} - \frac{89590}{186381} a - \frac{4360}{20709}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{14024}{1677429} a^{11} - \frac{39178}{1677429} a^{10} + \frac{13372}{129033} a^{9} - \frac{57298}{559143} a^{8} + \frac{55967}{129033} a^{7} - \frac{616709}{1677429} a^{6} + \frac{15967}{14337} a^{5} - \frac{334675}{1677429} a^{4} + \frac{1863937}{1677429} a^{3} + \frac{1583515}{1677429} a^{2} + \frac{245413}{186381} a + \frac{20188}{20709} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 697.177965406 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6$ (as 12T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-3}) \), 3.1.255.1 x3, \(\Q(\sqrt{-3}, \sqrt{85})\), 6.0.16581375.1, 6.2.5527125.2 x3, 6.0.195075.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 siblings: data not computed

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$