# Properties

 Label 6.2.1679616.2 Degree $6$ Signature $[2, 2]$ Discriminant $2^{8}\cdot 3^{8}$ Root discriminant $10.90$ Ramified primes $2, 3$ Class number $1$ Class group Trivial Galois group $A_6$ (as 6T15)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 6, -6, 0, 3, -3, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 3*x^4 - 6*x^2 + 6*x - 2)

gp: K = bnfinit(x^6 - 3*x^5 + 3*x^4 - 6*x^2 + 6*x - 2, 1)

## Normalizeddefining polynomial

$$x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$$

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: $$1679616=2^{8}\cdot 3^{8}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $10.90$ 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: $2, 3$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

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

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: $$a^{5} - 2 a^{4} + a^{3} + a^{2} - 5 a + 1$$,  $$a^{5} - 3 a^{4} + 3 a^{3} - 5 a + 5$$,  $$a^{4} - 2 a^{3} + a^{2} + a - 5$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$12.2562739075$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$A_6$ (as 6T15):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A non-solvable group of order 360 The 7 conjugacy class representatives for $A_6$ Character table for $A_6$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: 6.2.6718464.1 Degree 6 sibling: 6.2.6718464.1 Degree 10 sibling: 10.2.11284439629824.1 Degree 15 siblings: Deg 15, 15.3.18953525353286467584.1 Degree 20 sibling: 20.4.127338577759142414150270976.1 Degree 30 siblings: data not computed Degree 36 sibling: data not computed Degree 40 sibling: data not computed Degree 45 sibling: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2} 2.4.8.7x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.2e10_3e8.6t15.1c1$5$ $2^{10} \cdot 3^{8}$ $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ $A_6$ (as 6T15) $1$ $1$
* 5.2e8_3e8.6t15.1c1$5$ $2^{8} \cdot 3^{8}$ $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ $A_6$ (as 6T15) $1$ $1$
8.2e18_3e16.36t555.1c1$8$ $2^{18} \cdot 3^{16}$ $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ $A_6$ (as 6T15) $1$ $0$
8.2e18_3e16.36t555.1c2$8$ $2^{18} \cdot 3^{16}$ $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ $A_6$ (as 6T15) $1$ $0$
9.2e18_3e16.10t26.1c1$9$ $2^{18} \cdot 3^{16}$ $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ $A_6$ (as 6T15) $1$ $1$
10.2e24_3e16.30t88.1c1$10$ $2^{24} \cdot 3^{16}$ $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ $A_6$ (as 6T15) $1$ $-2$

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 *.