# Properties

 Label 2.99.6t5.a.b Dimension $2$ Group $S_3\times C_3$ Conductor $99$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$99$$$$\medspace = 3^{2} \cdot 11$$ Artin stem field: Galois closure of 6.0.107811.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.99.6t1.a.b Projective image: $S_3$ Projective stem field: Galois closure of 3.1.891.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{4} - 2x^{3} + 3x^{2} + x + 1$$ x^6 - x^4 - 2*x^3 + 3*x^2 + x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$11 a + 1 + \left(a + 2\right)\cdot 17 + \left(15 a + 1\right)\cdot 17^{2} + 15 a\cdot 17^{3} + \left(5 a + 5\right)\cdot 17^{4} +O(17^{5})$$ 11*a + 1 + (a + 2)*17 + (15*a + 1)*17^2 + 15*a*17^3 + (5*a + 5)*17^4+O(17^5) $r_{ 2 }$ $=$ $$6 a + 12 + \left(15 a + 9\right)\cdot 17 + \left(a + 14\right)\cdot 17^{2} + a\cdot 17^{3} + \left(11 a + 12\right)\cdot 17^{4} +O(17^{5})$$ 6*a + 12 + (15*a + 9)*17 + (a + 14)*17^2 + a*17^3 + (11*a + 12)*17^4+O(17^5) $r_{ 3 }$ $=$ $$14 a + 6 + \left(13 a + 1\right)\cdot 17 + \left(2 a + 15\right)\cdot 17^{2} + 14\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})$$ 14*a + 6 + (13*a + 1)*17 + (2*a + 15)*17^2 + 14*17^3 + 14*17^4+O(17^5) $r_{ 4 }$ $=$ $$3 a + 13 + \left(12 a + 13\right)\cdot 17 + \left(4 a + 11\right)\cdot 17^{2} + \left(a + 4\right)\cdot 17^{3} + \left(11 a + 14\right)\cdot 17^{4} +O(17^{5})$$ 3*a + 13 + (12*a + 13)*17 + (4*a + 11)*17^2 + (a + 4)*17^3 + (11*a + 14)*17^4+O(17^5) $r_{ 5 }$ $=$ $$3 a + 3 + \left(3 a + 1\right)\cdot 17 + \left(14 a + 4\right)\cdot 17^{2} + \left(16 a + 12\right)\cdot 17^{3} + \left(16 a + 14\right)\cdot 17^{4} +O(17^{5})$$ 3*a + 3 + (3*a + 1)*17 + (14*a + 4)*17^2 + (16*a + 12)*17^3 + (16*a + 14)*17^4+O(17^5) $r_{ 6 }$ $=$ $$14 a + 16 + \left(4 a + 5\right)\cdot 17 + \left(12 a + 4\right)\cdot 17^{2} + \left(15 a + 1\right)\cdot 17^{3} + \left(5 a + 7\right)\cdot 17^{4} +O(17^{5})$$ 14*a + 16 + (4*a + 5)*17 + (12*a + 4)*17^2 + (15*a + 1)*17^3 + (5*a + 7)*17^4+O(17^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,6,4,3,5,2)$ $(1,4,5)$ $(2,3,6)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,3)(2,4)(5,6)$ $0$ $1$ $3$ $(1,4,5)(2,6,3)$ $-2 \zeta_{3} - 2$ $1$ $3$ $(1,5,4)(2,3,6)$ $2 \zeta_{3}$ $2$ $3$ $(1,4,5)$ $-\zeta_{3}$ $2$ $3$ $(1,5,4)$ $\zeta_{3} + 1$ $2$ $3$ $(1,4,5)(2,3,6)$ $-1$ $3$ $6$ $(1,6,4,3,5,2)$ $0$ $3$ $6$ $(1,2,5,3,4,6)$ $0$

The blue line marks the conjugacy class containing complex conjugation.