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Properties

Label 12.0.2435758881214464.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 13^{8}$
Root discriminant $19.15$
Ramified primes $2, 3, 13$
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, -160, 200, -404, 489, -86, -38, 68, -33, 8, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 2*x^10 + 8*x^9 - 33*x^8 + 68*x^7 - 38*x^6 - 86*x^5 + 489*x^4 - 404*x^3 + 200*x^2 - 160*x + 64)
 
gp: K = bnfinit(x^12 - 2*x^11 + 2*x^10 + 8*x^9 - 33*x^8 + 68*x^7 - 38*x^6 - 86*x^5 + 489*x^4 - 404*x^3 + 200*x^2 - 160*x + 64, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 2 x^{10} + 8 x^{9} - 33 x^{8} + 68 x^{7} - 38 x^{6} - 86 x^{5} + 489 x^{4} - 404 x^{3} + 200 x^{2} - 160 x + 64 \)

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:  \(2435758881214464=2^{12}\cdot 3^{6}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.15$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{24} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{3} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{9} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{5}{24} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{3} a$, $\frac{1}{5976} a^{10} + \frac{43}{2988} a^{9} - \frac{37}{5976} a^{8} - \frac{113}{1992} a^{7} - \frac{1141}{5976} a^{6} - \frac{113}{5976} a^{5} + \frac{335}{2988} a^{4} - \frac{779}{1992} a^{3} + \frac{299}{2988} a^{2} - \frac{91}{1494} a + \frac{160}{747}$, $\frac{1}{6179184} a^{11} - \frac{217}{3089592} a^{10} + \frac{6749}{514932} a^{9} - \frac{59647}{3089592} a^{8} - \frac{909505}{6179184} a^{7} + \frac{50087}{280872} a^{6} + \frac{3763}{1029864} a^{5} + \frac{184037}{1544796} a^{4} - \frac{2400389}{6179184} a^{3} + \frac{727435}{1544796} a^{2} - \frac{441}{913} a + \frac{169801}{386199}$

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:  \( -\frac{5047}{3089592} a^{11} - \frac{7045}{1544796} a^{10} + \frac{4645}{772398} a^{9} - \frac{19363}{1029864} a^{8} - \frac{15803}{772398} a^{7} + \frac{13571}{140436} a^{6} - \frac{914803}{3089592} a^{5} + \frac{19031}{257466} a^{4} + \frac{10081}{386199} a^{3} - \frac{7765223}{3089592} a^{2} + \frac{184}{8217} a - \frac{1088}{42911} \) (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:  \( 2276.16399152 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 3.1.676.1 x3, \(\Q(\zeta_{12})\), 6.0.1827904.2, 6.2.49353408.1 x3, 6.0.12338352.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$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$$[2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$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}$
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$