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Properties

Label 6.2.10368000.1
Degree $6$
Signature $[2, 2]$
Discriminant $2^{10}\cdot 3^{4}\cdot 5^{3}$
Root discriminant $14.77$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $\PGL(2,5)$ (as 6T14)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(10368000=2^{10}\cdot 3^{4}\cdot 5^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.77$
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:  $2, 3, 5$
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 \(a\))

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

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{1}{3} a^{4} - \frac{1}{3} a - 1 \),  \( \frac{1}{3} a^{5} - \frac{2}{3} a^{4} + a^{3} - \frac{4}{3} a^{2} + \frac{2}{3} a - 1 \),  \( \frac{1}{3} a^{4} - a^{3} + a^{2} - \frac{1}{3} a - 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21.9198018435 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 6T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $\PGL(2,5)$
Character table for $\PGL(2,5)$

Intermediate fields

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

Sibling algebras

Twin sextic algebra: \(\Q\) $\times$ 5.1.103680.1
Degree 5 sibling: 5.1.103680.1
Degree 10 siblings: 10.2.214990848000.1, 10.2.1343692800000.3
Degree 12 sibling: 12.4.107495424000000.1
Degree 15 sibling: 15.3.2229025112064000000.1
Degree 20 siblings: 20.4.28888165452349440000000000.1, 20.0.369768517790072832000000000.1, Deg 20
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.10.4$x^{6} + 2 x^{5} + 2 x^{4} + 6$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
4.2e8_3e4_5e3.10t12.7c1$4$ $ 2^{8} \cdot 3^{4} \cdot 5^{3}$ $x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} - 4 x - 6$ $\PGL(2,5)$ (as 6T14) $1$ $0$
4.2e8_3e4_5.5t5.2c1$4$ $ 2^{8} \cdot 3^{4} \cdot 5 $ $x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} - 4 x - 6$ $\PGL(2,5)$ (as 6T14) $1$ $0$
5.2e10_3e4_5e2.10t13.2c1$5$ $ 2^{10} \cdot 3^{4} \cdot 5^{2}$ $x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} - 4 x - 6$ $\PGL(2,5)$ (as 6T14) $1$ $1$
* 5.2e10_3e4_5e3.6t14.2c1$5$ $ 2^{10} \cdot 3^{4} \cdot 5^{3}$ $x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} - 4 x - 6$ $\PGL(2,5)$ (as 6T14) $1$ $1$
6.2e16_3e4_5e3.20t35.2c1$6$ $ 2^{16} \cdot 3^{4} \cdot 5^{3}$ $x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} - 4 x - 6$ $\PGL(2,5)$ (as 6T14) $1$ $-2$

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 *.