# Properties

 Label 2.163.8t12.1c2 Dimension 2 Group $\SL(2,3)$ Conductor $163$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\SL(2,3)$ Conductor: $163$ Artin number field: Splitting field of $f=x^{8} - x^{7} + x^{6} - 4 x^{5} + 5 x^{4} - 8 x^{3} + 4 x^{2} - 8 x + 16$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 8T12 Parity: Even Determinant: 1.163.3t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{3} + x + 40$
Roots: \begin{aligned} r_{ 1 } &= 54114049 a^{2} - 25115108 a - 39218542 +O\left(43^{ 5 }\right) \\ r_{ 2 } &= -2097671 a^{2} + 63781279 a - 51373710 +O\left(43^{ 5 }\right) \\ r_{ 3 } &= -53911619 a^{2} - 30911808 a + 61092101 +O\left(43^{ 5 }\right) \\ r_{ 4 } &= 154124 +O\left(43^{ 5 }\right) \\ r_{ 5 } &= -65369940 +O\left(43^{ 5 }\right) \\ r_{ 6 } &= -44417676 a^{2} - 12835624 a - 6900730 +O\left(43^{ 5 }\right) \\ r_{ 7 } &= -9696373 a^{2} + 37950732 a - 32756009 +O\left(43^{ 5 }\right) \\ r_{ 8 } &= 56009290 a^{2} - 32869471 a - 12635736 +O\left(43^{ 5 }\right) \\ \end{aligned}

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(1,8)(2,7)(3,6)(4,5)$ $(1,6,7)(2,8,3)$ $(1,7,8,2)(3,5,6,4)$ $(1,4,8,5)(2,3,7,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $4$ $3$ $(2,6,4)(3,5,7)$ $\zeta_{3} + 1$ $4$ $3$ $(2,4,6)(3,7,5)$ $-\zeta_{3}$ $6$ $4$ $(1,7,8,2)(3,5,6,4)$ $0$ $4$ $6$ $(1,3,7,8,6,2)(4,5)$ $-\zeta_{3} - 1$ $4$ $6$ $(1,2,6,8,7,3)(4,5)$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.