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Properties

Label 12.0.10946861024053504.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 11^{6}\cdot 17^{6}$
Root discriminant $21.71$
Ramified primes $2, 11, 17$
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![4096, 1024, 3584, 1984, 864, 372, 101, -93, 54, -31, 14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 14*x^10 - 31*x^9 + 54*x^8 - 93*x^7 + 101*x^6 + 372*x^5 + 864*x^4 + 1984*x^3 + 3584*x^2 + 1024*x + 4096)
 
gp: K = bnfinit(x^12 - x^11 + 14*x^10 - 31*x^9 + 54*x^8 - 93*x^7 + 101*x^6 + 372*x^5 + 864*x^4 + 1984*x^3 + 3584*x^2 + 1024*x + 4096, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 14 x^{10} - 31 x^{9} + 54 x^{8} - 93 x^{7} + 101 x^{6} + 372 x^{5} + 864 x^{4} + 1984 x^{3} + 3584 x^{2} + 1024 x + 4096 \)

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:  \(10946861024053504=2^{8}\cdot 11^{6}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.71$
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:  $2, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{3}{8} a^{4} + \frac{3}{16} a^{3} + \frac{5}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} - \frac{15}{64} a^{6} + \frac{11}{32} a^{5} + \frac{19}{64} a^{4} + \frac{5}{64} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{20626688} a^{10} - \frac{143609}{20626688} a^{9} + \frac{129251}{10313344} a^{8} - \frac{1614831}{20626688} a^{7} + \frac{4394527}{10313344} a^{6} + \frac{1754387}{20626688} a^{5} + \frac{8111485}{20626688} a^{4} + \frac{2484279}{5156672} a^{3} - \frac{16783}{322292} a^{2} + \frac{150721}{322292} a - \frac{4}{80573}$, $\frac{1}{5032911872} a^{11} + \frac{99}{5032911872} a^{10} - \frac{9275339}{2516455936} a^{9} - \frac{55619559}{5032911872} a^{8} + \frac{201136397}{2516455936} a^{7} + \frac{2451354491}{5032911872} a^{6} + \frac{1508142609}{5032911872} a^{5} - \frac{119103515}{629113984} a^{4} - \frac{22098353}{314556992} a^{3} - \frac{4340549}{78639248} a^{2} - \frac{1105573}{9829906} a + \frac{1387179}{4914953}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 446.414194347 \)
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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{17}) \), 3.1.44.1 x3, \(\Q(\sqrt{-11}, \sqrt{17})\), 6.0.21296.1, 6.0.104627248.1 x3, 6.2.9511568.1 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$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}$