# Properties

 Label 6.6.170067681.1 Degree $6$ Signature $[6, 0]$ Discriminant $3^{8}\cdot 7^{2}\cdot 23^{2}$ Root discriminant $23.54$ Ramified primes $3, 7, 23$ Class number $1$ Class group Trivial Galois group $A_6$ (as 6T15)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 9*x^4 + 14*x^3 + 27*x^2 - 3*x - 10)

gp: K = bnfinit(x^6 - 3*x^5 - 9*x^4 + 14*x^3 + 27*x^2 - 3*x - 10, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, -3, 27, 14, -9, -3, 1]);

## Normalizeddefining polynomial

$$x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[6, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$170067681=3^{8}\cdot 7^{2}\cdot 23^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $23.54$ 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: $3, 7, 23$ 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 $$a$$)

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

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: $5$ 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: $$a + 1$$,  $$\frac{2}{3} a^{5} - \frac{8}{3} a^{4} - 3 a^{3} + \frac{35}{3} a^{2} + 4 a - \frac{13}{3}$$,  $$\frac{1}{3} a^{5} - \frac{4}{3} a^{4} - 2 a^{3} + \frac{19}{3} a^{2} + 4 a - \frac{11}{3}$$,  $$a^{2} - a - 3$$,  $$\frac{1}{3} a^{5} - \frac{5}{3} a^{4} - \frac{1}{3} a^{3} + \frac{20}{3} a^{2} + \frac{4}{3} a - 3$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$277.555419546$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$A_6$ (as 6T15):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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: 6.6.4408324359201.1 Degree 6 sibling: 6.6.4408324359201.1 Degree 10 sibling: Deg 10 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 ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

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
$3$3.6.8.2$x^{6} + 6 x^{5} + 9 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4} 77.3.2.2x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3} 23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.3.2.1x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
5.4408324359201.6t15.a.a$5$ $3^{8} \cdot 7^{4} \cdot 23^{4}$ $x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$ $A_6$ (as 6T15) $1$ $5$
* 5.170067681.6t15.a.a$5$ $3^{8} \cdot 7^{2} \cdot 23^{2}$ $x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$ $A_6$ (as 6T15) $1$ $5$
8.749713500865125082881.36t555.a.a$8$ $3^{16} \cdot 7^{6} \cdot 23^{6}$ $x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$ $A_6$ (as 6T15) $1$ $8$
8.749713500865125082881.36t555.a.b$8$ $3^{16} \cdot 7^{6} \cdot 23^{6}$ $x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$ $A_6$ (as 6T15) $1$ $8$
9.749713500865125082881.10t26.a.a$9$ $3^{16} \cdot 7^{6} \cdot 23^{6}$ $x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$ $A_6$ (as 6T15) $1$ $9$
10.749713500865125082881.30t88.a.a$10$ $3^{16} \cdot 7^{6} \cdot 23^{6}$ $x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10$ $A_6$ (as 6T15) $1$ $10$

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 *.