Properties

Label 6.2.51757.1
Degree $6$
Signature $[2, 2]$
Discriminant $73\cdot 709$
Root discriminant $6.10$
Ramified primes $73, 709$
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, -2, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^3 - 2*x^2 + x + 1)
gp: K = bnfinit(x^6 - 2*x^3 - 2*x^2 + x + 1, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut -\mathstrut 2 x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(51757=73\cdot 709\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $6.10$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $73, 709$
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:  \( a^{5} - 2 a^{2} - 2 a + 1 \),  \( a^{5} - a^{4} - 2 a^{2} + 2 \),  \( a^{5} + a^{3} - 2 a^{2} - 2 a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 0.7419673261 \)
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: 6.2.138645981295093.1
Degree 6 sibling: 6.2.138645981295093.1
Degree 10 sibling: 10.2.138645981295093.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
709Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.73_709.2t1.1c1$1$ $ 73 \cdot 709 $ $x^{2} - x - 12939$ $C_2$ (as 2T1) $1$ $1$
* 5.73_709.6t16.1c1$5$ $ 73 \cdot 709 $ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
5.73e2_709e2.12t183.1c1$5$ $ 73^{2} \cdot 709^{2}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
5.73e4_709e4.12t183.1c1$5$ $ 73^{4} \cdot 709^{4}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
5.73e3_709e3.6t16.1c1$5$ $ 73^{3} \cdot 709^{3}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
9.73e3_709e3.10t32.1c1$9$ $ 73^{3} \cdot 709^{3}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
9.73e6_709e6.20t145.1c1$9$ $ 73^{6} \cdot 709^{6}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
10.73e6_709e6.30t176.1c1$10$ $ 73^{6} \cdot 709^{6}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-2$
10.73e4_709e4.30t176.1c1$10$ $ 73^{4} \cdot 709^{4}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-2$
16.73e8_709e8.36t1252.1c1$16$ $ 73^{8} \cdot 709^{8}$ $x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ $S_6$ (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 *.