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Properties

Label 12.0.4469547301936929.2
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 19^{10}$
Root discriminant $20.15$
Ramified primes $3, 19$
Class number $3$
Class group $[3]$
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![64, -8, -39, 107, 10, -9, 31, -36, 27, -11, 10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64)
 
gp: K = bnfinit(x^12 - 3*x^11 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 10 x^{10} - 11 x^{9} + 27 x^{8} - 36 x^{7} + 31 x^{6} - 9 x^{5} + 10 x^{4} + 107 x^{3} - 39 x^{2} - 8 x + 64 \)

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:  \(4469547301936929=3^{6}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.15$
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:  $3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{12} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{12} a$, $\frac{1}{24} a^{8} - \frac{1}{24} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{4} - \frac{5}{12} a^{3} + \frac{7}{24} a^{2} + \frac{1}{8} a + \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{7} - \frac{1}{12} a^{6} - \frac{1}{12} a^{5} + \frac{5}{24} a^{3} - \frac{5}{12} a^{2} + \frac{1}{24} a + \frac{1}{3}$, $\frac{1}{360} a^{10} - \frac{7}{360} a^{9} + \frac{1}{360} a^{8} - \frac{11}{360} a^{7} + \frac{1}{90} a^{6} - \frac{5}{36} a^{5} - \frac{7}{360} a^{4} + \frac{11}{40} a^{3} + \frac{53}{360} a^{2} - \frac{1}{120} a + \frac{14}{45}$, $\frac{1}{16560} a^{11} + \frac{7}{8280} a^{10} + \frac{4}{1035} a^{9} + \frac{59}{3312} a^{8} + \frac{7}{4140} a^{7} + \frac{271}{4140} a^{6} - \frac{1597}{16560} a^{5} + \frac{457}{2760} a^{4} - \frac{1477}{4140} a^{3} + \frac{49}{368} a^{2} - \frac{247}{2070} a - \frac{57}{115}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{7}{5520} a^{11} - \frac{13}{1380} a^{10} + \frac{5}{184} a^{9} - \frac{213}{1840} a^{8} + \frac{113}{920} a^{7} - \frac{157}{460} a^{6} + \frac{3359}{5520} a^{5} - \frac{739}{1380} a^{4} + \frac{89}{184} a^{3} - \frac{3337}{5520} a^{2} - \frac{2123}{2760} a + \frac{116}{115} \) (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:  \( 2885.43572816 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{57}) \), 3.1.1083.1 x3, \(\Q(\sqrt{-3}, \sqrt{-19})\), 6.0.3518667.2, 6.0.22284891.1 x3, 6.2.66854673.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$