# Properties

 Label 2.163.8t12.a.b 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: $\SL(2,3)$ Parity: Even Determinant: 1.163.3t1.a.b

## 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:
 $r_{ 1 }$ $=$ $11 a^{2} + 31 a + 38 + \left(28 a^{2} + 39 a + 13\right)\cdot 43 + \left(26 a^{2} + 4 a + 31\right)\cdot 43^{2} + \left(35 a^{2} + 28 a + 22\right)\cdot 43^{3} + \left(15 a^{2} + 35 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $41 a^{2} + 24 a + 24 + \left(21 a^{2} + 17\right)\cdot 43 + \left(26 a^{2} + 9 a + 36\right)\cdot 43^{2} + \left(16 a^{2} + 28 a + 41\right)\cdot 43^{3} + \left(42 a^{2} + 18 a + 27\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $18 a^{2} + 32 a + 23 + \left(35 a^{2} + 37 a + 26\right)\cdot 43 + \left(39 a^{2} + 8 a + 16\right)\cdot 43^{2} + \left(9 a^{2} + 41 a + 37\right)\cdot 43^{3} + \left(27 a^{2} + 33 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $12 + 15\cdot 43 + 40\cdot 43^{2} + 43^{3} +O\left(43^{ 5 }\right)$ $r_{ 5 }$ $=$ $36 + 33\cdot 43 + 34\cdot 43^{2} + 37\cdot 43^{3} + 23\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 6 }$ $=$ $34 a^{2} + 5 a + 39 + \left(19 a^{2} + 3 a + 36\right)\cdot 43 + \left(14 a^{2} + 24 a + 8\right)\cdot 43^{2} + \left(10 a + 42\right)\cdot 43^{3} + \left(30 a^{2} + 39 a + 40\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 7 }$ $=$ $41 a^{2} + 7 a + 15 + \left(37 a^{2} + 20\right)\cdot 43 + \left(a^{2} + 14 a\right)\cdot 43^{2} + \left(7 a^{2} + 4 a + 18\right)\cdot 43^{3} + \left(40 a^{2} + 11 a + 33\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 8 }$ $=$ $27 a^{2} + 30 a + 29 + \left(28 a^{2} + 4 a + 7\right)\cdot 43 + \left(19 a^{2} + 25 a + 3\right)\cdot 43^{2} + \left(16 a^{2} + 16 a + 13\right)\cdot 43^{3} + \left(16 a^{2} + 33 a + 39\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$

### 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.