# Properties

 Label 2.3e2_239.24t22.2c1 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $3^{2} \cdot 239$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $2151= 3^{2} \cdot 239$ Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 4 x^{6} + 2 x^{5} - 8 x^{4} + 8 x^{3} + 10 x^{2} - 13 x - 5$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.239.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{2} + 42 x + 3$
Roots:
 $r_{ 1 }$ $=$ $29 a + 24 + \left(38 a + 22\right)\cdot 43 + \left(33 a + 31\right)\cdot 43^{2} + \left(20 a + 41\right)\cdot 43^{3} + \left(3 a + 19\right)\cdot 43^{4} + \left(42 a + 22\right)\cdot 43^{5} + \left(38 a + 41\right)\cdot 43^{6} + \left(17 a + 38\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ $r_{ 2 }$ $=$ $9 + 34\cdot 43 + 23\cdot 43^{2} + 16\cdot 43^{3} + 16\cdot 43^{5} + 3\cdot 43^{6} + 12\cdot 43^{7} +O\left(43^{ 8 }\right)$ $r_{ 3 }$ $=$ $35 + 8\cdot 43 + 19\cdot 43^{2} + 26\cdot 43^{3} + 42\cdot 43^{4} + 26\cdot 43^{5} + 39\cdot 43^{6} + 30\cdot 43^{7} +O\left(43^{ 8 }\right)$ $r_{ 4 }$ $=$ $22 a + 11 + \left(18 a + 23\right)\cdot 43 + \left(26 a + 17\right)\cdot 43^{2} + \left(23 a + 1\right)\cdot 43^{3} + \left(7 a + 8\right)\cdot 43^{4} + \left(23 a + 35\right)\cdot 43^{5} + \left(11 a + 5\right)\cdot 43^{6} + \left(40 a + 7\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ $r_{ 5 }$ $=$ $21 a + 33 + \left(24 a + 19\right)\cdot 43 + \left(16 a + 25\right)\cdot 43^{2} + \left(19 a + 41\right)\cdot 43^{3} + \left(35 a + 34\right)\cdot 43^{4} + \left(19 a + 7\right)\cdot 43^{5} + \left(31 a + 37\right)\cdot 43^{6} + \left(2 a + 35\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ $r_{ 6 }$ $=$ $14 a + 10 + \left(4 a + 32\right)\cdot 43 + \left(9 a + 26\right)\cdot 43^{2} + \left(22 a + 28\right)\cdot 43^{3} + \left(39 a + 2\right)\cdot 43^{4} + 18\cdot 43^{5} + \left(4 a + 38\right)\cdot 43^{6} + \left(25 a + 17\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ $r_{ 7 }$ $=$ $29 a + 34 + \left(38 a + 10\right)\cdot 43 + \left(33 a + 16\right)\cdot 43^{2} + \left(20 a + 14\right)\cdot 43^{3} + \left(3 a + 40\right)\cdot 43^{4} + \left(42 a + 24\right)\cdot 43^{5} + \left(38 a + 4\right)\cdot 43^{6} + \left(17 a + 25\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ $r_{ 8 }$ $=$ $14 a + 20 + \left(4 a + 20\right)\cdot 43 + \left(9 a + 11\right)\cdot 43^{2} + \left(22 a + 1\right)\cdot 43^{3} + \left(39 a + 23\right)\cdot 43^{4} + 20\cdot 43^{5} + \left(4 a + 1\right)\cdot 43^{6} + \left(25 a + 4\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,8)(2,3)(4,5)(6,7)$ $-2$ $12$ $2$ $(1,8)(4,7)(5,6)$ $0$ $8$ $3$ $(1,4,2)(3,8,5)$ $-1$ $6$ $4$ $(1,6,8,7)(2,5,3,4)$ $0$ $8$ $6$ $(1,3,4,8,2,5)(6,7)$ $1$ $6$ $8$ $(1,7,2,5,8,6,3,4)$ $-\zeta_{8}^{3} - \zeta_{8}$ $6$ $8$ $(1,6,2,4,8,7,3,5)$ $\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.