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Properties

Label 12.0.44047357079191552.4
Degree $12$
Signature $[0, 6]$
Discriminant $2^{31}\cdot 29^{5}$
Root discriminant $24.38$
Ramified primes $2, 29$
Class number $1$
Class group Trivial
Galois group $D_6:D_6$ (as 12T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50, -336, 1018, -1720, 1721, -1012, 400, -180, 98, -36, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 36*x^9 + 98*x^8 - 180*x^7 + 400*x^6 - 1012*x^5 + 1721*x^4 - 1720*x^3 + 1018*x^2 - 336*x + 50)
 
gp: K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 36*x^9 + 98*x^8 - 180*x^7 + 400*x^6 - 1012*x^5 + 1721*x^4 - 1720*x^3 + 1018*x^2 - 336*x + 50, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 10 x^{10} - 36 x^{9} + 98 x^{8} - 180 x^{7} + 400 x^{6} - 1012 x^{5} + 1721 x^{4} - 1720 x^{3} + 1018 x^{2} - 336 x + 50 \)

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:  \(44047357079191552=2^{31}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.38$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5781633299} a^{11} + \frac{61666385}{5781633299} a^{10} + \frac{1179149103}{5781633299} a^{9} - \frac{18225713}{444741023} a^{8} + \frac{3535921}{59604467} a^{7} - \frac{2435018146}{5781633299} a^{6} - \frac{3173514}{444741023} a^{5} - \frac{2615747639}{5781633299} a^{4} + \frac{1260285717}{5781633299} a^{3} - \frac{566290511}{5781633299} a^{2} - \frac{2685840565}{5781633299} a + \frac{1581918341}{5781633299}$

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:  \( -\frac{1084286}{4584959} a^{11} + \frac{4302073}{4584959} a^{10} - \frac{10395690}{4584959} a^{9} + \frac{37674385}{4584959} a^{8} - \frac{102860866}{4584959} a^{7} + \frac{182728700}{4584959} a^{6} - \frac{404978562}{4584959} a^{5} + \frac{1049374675}{4584959} a^{4} - \frac{1743517760}{4584959} a^{3} + \frac{1572749373}{4584959} a^{2} - \frac{731373112}{4584959} a + \frac{146226513}{4584959} \) (order $4$)
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:  \( 19554.5458202 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6:D_6$ (as 12T81):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $D_6:D_6$
Character table for $D_6:D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.14848.1, 6.0.861184.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.4.9.4$x^{4} + 2 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.22.102$x^{8} + 8 x^{7} + 16 x^{5} + 144$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.5.1$x^{6} - 29$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$