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Properties

Label 12.0.11057373810786304.4
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 59^{6}$
Root discriminant $21.73$
Ramified primes $2, 59$
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![2564, 1872, -1468, -788, 391, -130, 273, -214, 82, -30, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 17*x^10 - 30*x^9 + 82*x^8 - 214*x^7 + 273*x^6 - 130*x^5 + 391*x^4 - 788*x^3 - 1468*x^2 + 1872*x + 2564)
 
gp: K = bnfinit(x^12 - 6*x^11 + 17*x^10 - 30*x^9 + 82*x^8 - 214*x^7 + 273*x^6 - 130*x^5 + 391*x^4 - 788*x^3 - 1468*x^2 + 1872*x + 2564, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 17 x^{10} - 30 x^{9} + 82 x^{8} - 214 x^{7} + 273 x^{6} - 130 x^{5} + 391 x^{4} - 788 x^{3} - 1468 x^{2} + 1872 x + 2564 \)

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:  \(11057373810786304=2^{18}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.73$
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:  $2, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{2470508} a^{10} - \frac{5}{2470508} a^{9} + \frac{135793}{2470508} a^{8} - \frac{271571}{1235254} a^{7} - \frac{226279}{1235254} a^{6} + \frac{394071}{1235254} a^{5} + \frac{956101}{2470508} a^{4} - \frac{565417}{2470508} a^{3} - \frac{469529}{2470508} a^{2} + \frac{75307}{1235254} a + \frac{171909}{1235254}$, $\frac{1}{8743127812} a^{11} + \frac{441}{2185781953} a^{10} - \frac{208076825}{4371563906} a^{9} - \frac{72844587}{8743127812} a^{8} + \frac{133964273}{2185781953} a^{7} - \frac{416116190}{2185781953} a^{6} + \frac{3655694119}{8743127812} a^{5} + \frac{641951898}{2185781953} a^{4} + \frac{212738386}{2185781953} a^{3} - \frac{1792709053}{8743127812} a^{2} + \frac{106223124}{2185781953} a + \frac{1467716289}{4371563906}$

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:  \( 762.546793808 \)
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

\(\Q(\sqrt{-59}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-118}) \), 3.1.59.1 x3, \(\Q(\sqrt{2}, \sqrt{-59})\), 6.0.205379.1, 6.2.1782272.1 x3, 6.0.105154048.1 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ R

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$59$59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$