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Properties

Label 12.0.7652750400000000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{14}\cdot 5^{8}$
Root discriminant $21.07$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
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![676, -156, -354, 950, -285, -276, 191, -126, 90, -50, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 191*x^6 - 276*x^5 - 285*x^4 + 950*x^3 - 354*x^2 - 156*x + 676)
 
gp: K = bnfinit(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 191*x^6 - 276*x^5 - 285*x^4 + 950*x^3 - 354*x^2 - 156*x + 676, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} + 191 x^{6} - 276 x^{5} - 285 x^{4} + 950 x^{3} - 354 x^{2} - 156 x + 676 \)

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:  \(7652750400000000=2^{12}\cdot 3^{14}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.07$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{7}{15} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{4}{15}$, $\frac{1}{15} a^{7} - \frac{1}{5} a^{5} - \frac{4}{15} a^{4} + \frac{2}{15} a + \frac{1}{5}$, $\frac{1}{30} a^{8} + \frac{1}{15} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{6} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{390} a^{9} + \frac{1}{195} a^{8} + \frac{2}{195} a^{7} - \frac{1}{65} a^{6} - \frac{58}{195} a^{5} - \frac{77}{195} a^{4} + \frac{109}{390} a^{3} + \frac{83}{195} a^{2} + \frac{49}{195} a - \frac{1}{3}$, $\frac{1}{1170} a^{10} + \frac{1}{1170} a^{9} + \frac{1}{585} a^{8} + \frac{8}{585} a^{7} - \frac{1}{195} a^{6} + \frac{176}{585} a^{5} + \frac{131}{390} a^{4} + \frac{499}{1170} a^{3} - \frac{112}{585} a^{2} - \frac{49}{585} a - \frac{8}{45}$, $\frac{1}{141570} a^{11} + \frac{1}{2574} a^{10} - \frac{133}{141570} a^{9} + \frac{1033}{141570} a^{8} + \frac{212}{23595} a^{7} - \frac{1681}{70785} a^{6} - \frac{20623}{47190} a^{5} + \frac{66739}{141570} a^{4} - \frac{67043}{141570} a^{3} + \frac{18793}{141570} a^{2} + \frac{7879}{70785} a + \frac{53}{363}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{189}{15730} a^{11} - \frac{289}{4290} a^{10} + \frac{91}{363} a^{9} - \frac{29453}{47190} a^{8} + \frac{6008}{4719} a^{7} - \frac{3668}{1815} a^{6} + \frac{165641}{47190} a^{5} - \frac{46037}{9438} a^{4} + \frac{38}{363} a^{3} + \frac{227609}{47190} a^{2} - \frac{47932}{7865} a + \frac{1062}{605} \) (order $12$)
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:  \( 15293.392846 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), 3.1.675.1 x3, \(\Q(\zeta_{12})\), 6.0.1366875.1, 6.2.87480000.1 x3, 6.0.29160000.1 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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$