Properties

Label 6.2.25920000.1
Degree $6$
Signature $[2, 2]$
Discriminant $2^{9}\cdot 3^{4}\cdot 5^{4}$
Root discriminant $17.20$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_6$ (as 6T16)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 12*x^3 - 14*x^2 + 8*x - 4)
 
gp: K = bnfinit(x^6 - 2*x^5 - 4*x^4 + 12*x^3 - 14*x^2 + 8*x - 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 8, -14, 12, -4, -2, 1]);
 

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4 \)

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:  \(25920000=2^{9}\cdot 3^{4}\cdot 5^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.20$
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:  $2, 3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

Trivial group, which has order $1$

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:  \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 13 a^{2} - 9 a + 5 \),  \( 2 a^{5} - 4 a^{4} - 9 a^{3} + 26 a^{2} - 23 a + 3 \),  \( \frac{1}{2} a^{5} - 2 a^{4} - 2 a^{3} + 12 a^{2} - 8 a + 5 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 129.485809974 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$S_6$ (as 6T16):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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: 6.2.1036800.1
Degree 6 sibling: 6.2.1036800.1
Degree 10 sibling: 10.2.3359232000000.1
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 R R ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.8.2t1.a.a$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 5.25920000.6t16.a.a$5$ $ 2^{9} \cdot 3^{4} \cdot 5^{4}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $1$
5.8294400.12t183.a.a$5$ $ 2^{12} \cdot 3^{4} \cdot 5^{2}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $1$
5.207360000.12t183.a.a$5$ $ 2^{12} \cdot 3^{4} \cdot 5^{4}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $1$
5.1036800.6t16.a.a$5$ $ 2^{9} \cdot 3^{4} \cdot 5^{2}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $1$
9.3359232000000.10t32.a.a$9$ $ 2^{15} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $1$
9.1719926784000000.20t145.a.a$9$ $ 2^{24} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $1$
10.1719926784000000.30t164.a.a$10$ $ 2^{24} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $-2$
10.1719926784000000.30t164.b.a$10$ $ 2^{24} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (as 6T16) $1$ $-2$
16.8916100448256000000000000.36t1252.a.a$16$ $ 2^{36} \cdot 3^{12} \cdot 5^{12}$ $x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 14 x^{2} + 8 x - 4$ $S_6$ (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 *.