Properties

Label 6.2.51681125.1
Degree $6$
Signature $[2, 2]$
Discriminant $5^{3}\cdot 643^{2}$
Root discriminant $19.30$
Ramified primes $5, 643$
Class number $4$
Class group $[4]$
Galois group $S_4\times C_2$ (as 6T11)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 4*x^4 - 22*x^3 + 4*x^2 - x - 1)
 
gp: K = bnfinit(x^6 - x^5 - 4*x^4 - 22*x^3 + 4*x^2 - x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 4, -22, -4, -1, 1]);
 

Normalized defining polynomial

\( x^{6} - x^{5} - 4 x^{4} - 22 x^{3} + 4 x^{2} - x - 1 \)

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

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(51681125=5^{3}\cdot 643^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.30$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 643$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{10} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{1}{2}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{4}$, which has order $4$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{5} - a^{4} - 4 a^{3} - 22 a^{2} + 4 a - 1 \),  \( \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{3}{5} a^{3} - \frac{19}{5} a^{2} + \frac{26}{5} a + \frac{8}{5} \),  \( \frac{11}{5} a^{5} - \frac{8}{5} a^{4} - \frac{48}{5} a^{3} - \frac{256}{5} a^{2} - \frac{24}{5} a + \frac{12}{5} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 51.418342098 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2\times S_4$ (as 6T11):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

3.1.643.1

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

Sibling algebras

Twin sextic algebra: 4.2.643.1 $\times$ \(\Q(\sqrt{-3215}) \)
Degree 6 sibling: 6.0.33230963375.2
Degree 8 siblings: 8.4.258405625.2, 8.0.106837547250625.3
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 16 sibling: Deg 16
Degree 24 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
643Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.a.a$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.643.2t1.a.a$1$ $ 643 $ $x^{2} - x + 161$ $C_2$ (as 2T1) $1$ $-1$
1.3215.2t1.a.a$1$ $ 5 \cdot 643 $ $x^{2} - x + 804$ $C_2$ (as 2T1) $1$ $-1$
2.16075.6t3.b.a$2$ $ 5^{2} \cdot 643 $ $x^{6} - 16 x^{4} - 55 x^{3} + 64 x^{2} + 440 x + 1560$ $D_{6}$ (as 6T3) $1$ $0$
* 2.643.3t2.a.a$2$ $ 643 $ $x^{3} - 2 x - 5$ $S_3$ (as 3T2) $1$ $0$
3.643.4t5.a.a$3$ $ 643 $ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $1$
3.51681125.6t11.a.a$3$ $ 5^{3} \cdot 643^{2}$ $x^{6} - x^{5} - 4 x^{4} - 22 x^{3} + 4 x^{2} - x - 1$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 3.80375.6t11.a.a$3$ $ 5^{3} \cdot 643 $ $x^{6} - x^{5} - 4 x^{4} - 22 x^{3} + 4 x^{2} - x - 1$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.413449.6t8.a.a$3$ $ 643^{2}$ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $-1$

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 *.