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Properties

Label 6.6.967458816.1
Degree $6$
Signature $[6, 0]$
Discriminant $2^{14}\cdot 3^{10}$
Root discriminant $31.45$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_4$ (as 6T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 0, 30, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 12*x^4 + 30*x^2 - 16)
 
gp: K = bnfinit(x^6 - 12*x^4 + 30*x^2 - 16, 1)
 

Normalized defining polynomial

\( x^{6} - 12 x^{4} + 30 x^{2} - 16 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(967458816=2^{14}\cdot 3^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.45$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{1}{4} a^{5} - 3 a^{3} + \frac{13}{2} a - 1 \),  \( \frac{1}{4} a^{5} - 3 a^{3} + \frac{13}{2} a + 1 \),  \( a^{5} - a^{4} - 10 a^{3} + 8 a^{2} + 18 a - 15 \),  \( \frac{1}{4} a^{5} - 2 a^{3} + \frac{5}{2} a - 1 \),  \( \frac{1}{4} a^{5} - 2 a^{3} + \frac{5}{2} a + 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1719.18270049 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_4$ (as 6T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

3.3.1944.1

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

Sibling algebras

Galois closure: data not computed
Twin sextic algebra: 4.4.497664.1 $\times$ \(\Q(\sqrt{6}) \)
Degree 4 sibling: 4.4.497664.1
Degree 6 sibling: 6.6.362797056.1
Degree 8 sibling: 8.8.2229025112064.1
Degree 12 siblings: 12.12.89853749822987698176.1, 12.12.8423789045905096704.2

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.11.14$x^{4} + 8 x + 10$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
$3$3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_3.2t1.1c1$1$ $ 2^{3} \cdot 3 $ $x^{2} - 6$ $C_2$ (as 2T1) $1$ $1$
* 2.2e3_3e5.3t2.1c1$2$ $ 2^{3} \cdot 3^{5}$ $x^{3} - 9 x - 6$ $S_3$ (as 3T2) $1$ $2$
3.2e8_3e6.6t8.8c1$3$ $ 2^{8} \cdot 3^{6}$ $x^{6} - 12 x^{4} + 30 x^{2} - 16$ $S_4$ (as 6T7) $1$ $3$
* 3.2e11_3e5.4t5.8c1$3$ $ 2^{11} \cdot 3^{5}$ $x^{6} - 12 x^{4} + 30 x^{2} - 16$ $S_4$ (as 6T7) $1$ $3$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.