# Properties

 Label 6.2.95004009.3 Degree $6$ Signature $[2, 2]$ Discriminant $3^{6}\cdot 19^{4}$ Root discriminant $21.36$ Ramified primes $3, 19$ 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![6, -9, 9, 8, -3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 3*x^4 + 8*x^3 + 9*x^2 - 9*x + 6)
gp: K = bnfinit(x^6 - 3*x^5 - 3*x^4 + 8*x^3 + 9*x^2 - 9*x + 6, 1)

## Normalizeddefining polynomial

$$x^{6}$$ $$\mathstrut -\mathstrut 3 x^{5}$$ $$\mathstrut -\mathstrut 3 x^{4}$$ $$\mathstrut +\mathstrut 8 x^{3}$$ $$\mathstrut +\mathstrut 9 x^{2}$$ $$\mathstrut -\mathstrut 9 x$$ $$\mathstrut +\mathstrut 6$$

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: $$95004009=3^{6}\cdot 19^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $21.36$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 19$ magma: PrimeDivisors(Discriminant(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 $$a$$)

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

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: $$a^{4} - a^{3} - 5 a^{2} - a + 7$$,  $$a^{2} - 3 a + 1$$,  $$\frac{104}{3} a^{5} - \frac{220}{3} a^{4} - \frac{562}{3} a^{3} + 153 a^{2} + 541 a + 73$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$201.334350946$$ 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: 6.2.7695324729.3 Degree 6 sibling: 6.2.7695324729.3 Degree 10 sibling: 10.2.731086699811838561.1 Degree 15 siblings: Deg 15, Deg 15 Degree 20 sibling: Deg 20 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 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0Trivial[\ ] 3.2.1.2x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2} 19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.3e10_19e4.6t15.2c1$5$ $3^{10} \cdot 19^{4}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $1$
* 5.3e6_19e4.6t15.2c1$5$ $3^{6} \cdot 19^{4}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $1$
8.3e16_19e6.36t555.2c1$8$ $3^{16} \cdot 19^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $0$
8.3e16_19e6.36t555.2c2$8$ $3^{16} \cdot 19^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $0$
9.3e16_19e8.10t26.2c1$9$ $3^{16} \cdot 19^{8}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $1$
10.3e18_19e8.30t88.2c1$10$ $3^{18} \cdot 19^{8}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (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 *.