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Properties

Label 6.6.2666432.1
Degree $6$
Signature $[6, 0]$
Discriminant $2^{6}\cdot 61\cdot 683$
Root discriminant $11.78$
Ramified primes $2, 61, 683$
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, -2, 7, 4, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 6*x^4 + 4*x^3 + 7*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^6 - 2*x^5 - 6*x^4 + 4*x^3 + 7*x^2 - 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2666432=2^{6}\cdot 61\cdot 683\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.78$
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:  $2, 61, 683$
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 \(a\))

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

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:  \( a^{5} - 2 a^{4} - 6 a^{3} + 4 a^{2} + 7 a - 2 \),  \( a^{4} - 3 a^{3} - 2 a^{2} + 4 a \),  \( a^{5} - 2 a^{4} - 6 a^{3} + 4 a^{2} + 6 a - 2 \),  \( a + 1 \),  \( a^{4} - 3 a^{3} - 2 a^{2} + 4 a - 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13.0212252837 \)
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: 6.6.18513630059839232.1
Degree 6 sibling: 6.6.18513630059839232.1
Degree 10 sibling: 10.10.296218080957427712.1
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 R ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
61Data not computed
683Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_61_683.2t1.1c1$1$ $ 2^{2} \cdot 61 \cdot 683 $ $x^{2} - 41663$ $C_2$ (as 2T1) $1$ $1$
* 5.2e6_61_683.6t16.1c1$5$ $ 2^{6} \cdot 61 \cdot 683 $ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $5$
5.2e8_61e2_683e2.12t183.1c1$5$ $ 2^{8} \cdot 61^{2} \cdot 683^{2}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $5$
5.2e10_61e4_683e4.12t183.1c1$5$ $ 2^{10} \cdot 61^{4} \cdot 683^{4}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $5$
5.2e8_61e3_683e3.6t16.1c1$5$ $ 2^{8} \cdot 61^{3} \cdot 683^{3}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $5$
9.2e12_61e3_683e3.10t32.1c1$9$ $ 2^{12} \cdot 61^{3} \cdot 683^{3}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $9$
9.2e18_61e6_683e6.20t145.1c1$9$ $ 2^{18} \cdot 61^{6} \cdot 683^{6}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $9$
10.2e20_61e6_683e6.30t176.1c1$10$ $ 2^{20} \cdot 61^{6} \cdot 683^{6}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $10$
10.2e18_61e4_683e4.30t176.1c1$10$ $ 2^{18} \cdot 61^{4} \cdot 683^{4}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $10$
16.2e28_61e8_683e8.36t1252.1c1$16$ $ 2^{28} \cdot 61^{8} \cdot 683^{8}$ $x^{6} - 2 x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - 2 x - 1$ $S_6$ (as 6T16) $1$ $16$

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 *.