Properties

Label 6.2.41069.1
Degree $6$
Signature $[2, 2]$
Discriminant $7\cdot 5867$
Root discriminant $5.87$
Ramified primes $7, 5867$
Class number $1$
Class group Trivial
Galois group $S_6$ (as 6T16)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\( x^{6} - 2 x^{4} + 2 x^{2} - x - 1 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41069=7\cdot 5867\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $5.87$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 5867$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$

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:  $3$
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:  \( a^{2} - 1 \),  \( a^{5} - 2 a^{3} + 2 a - 1 \),  \( a^{5} - 2 a^{3} + 2 a \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 0.632541712933 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_6$ (as 6T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for $S_6$
Character table for $S_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling algebras

Twin sextic algebra: 6.2.69269552931509.1
Degree 6 sibling: 6.2.69269552931509.1
Degree 10 sibling: 10.2.69269552931509.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
5867Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7_5867.2t1.1c1$1$ $ 7 \cdot 5867 $ $x^{2} - x - 10267$ $C_2$ (as 2T1) $1$ $1$
* 5.7_5867.6t16.1c1$5$ $ 7 \cdot 5867 $ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $1$
5.7e2_5867e2.12t183.1c1$5$ $ 7^{2} \cdot 5867^{2}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $1$
5.7e4_5867e4.12t183.1c1$5$ $ 7^{4} \cdot 5867^{4}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $1$
5.7e3_5867e3.6t16.1c1$5$ $ 7^{3} \cdot 5867^{3}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $1$
9.7e3_5867e3.10t32.1c1$9$ $ 7^{3} \cdot 5867^{3}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $1$
9.7e6_5867e6.20t145.1c1$9$ $ 7^{6} \cdot 5867^{6}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $1$
10.7e6_5867e6.30t176.1c1$10$ $ 7^{6} \cdot 5867^{6}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $-2$
10.7e4_5867e4.30t176.1c1$10$ $ 7^{4} \cdot 5867^{4}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $-2$
16.7e8_5867e8.36t1252.1c1$16$ $ 7^{8} \cdot 5867^{8}$ $x^{6} - 2 x^{4} + 2 x^{2} - x - 1$ $S_6$ (as 6T16) $1$ $0$

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 *.