# Properties

 Label 2.2432.24t22.a.b Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{7} \cdot 19$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $2432= 2^{7} \cdot 19$ Artin number field: Splitting field of 8.2.28768731136.1 defined by $f= x^{8} - 12 x^{4} - 16 x^{2} - 76$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.19.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.2.4864.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $x^{2} + 58 x + 2$
Roots:
 $r_{ 1 }$ $=$ $33 a + 10 + \left(49 a + 29\right)\cdot 59 + \left(35 a + 35\right)\cdot 59^{2} + \left(23 a + 24\right)\cdot 59^{3} + \left(29 a + 10\right)\cdot 59^{4} + \left(4 a + 4\right)\cdot 59^{5} + \left(18 a + 5\right)\cdot 59^{6} + \left(17 a + 56\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 2 }$ $=$ $22 a + 48 + \left(4 a + 8\right)\cdot 59 + \left(38 a + 42\right)\cdot 59^{2} + \left(22 a + 7\right)\cdot 59^{3} + 22 a\cdot 59^{4} + \left(41 a + 20\right)\cdot 59^{5} + \left(35 a + 32\right)\cdot 59^{6} + \left(24 a + 5\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 3 }$ $=$ $56 + 8\cdot 59 + 58\cdot 59^{2} + 51\cdot 59^{3} + 25\cdot 59^{4} + 54\cdot 59^{5} + 23\cdot 59^{6} + 36\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 4 }$ $=$ $33 a + 16 + \left(49 a + 13\right)\cdot 59 + \left(35 a + 37\right)\cdot 59^{2} + \left(23 a + 46\right)\cdot 59^{3} + \left(29 a + 42\right)\cdot 59^{4} + \left(4 a + 20\right)\cdot 59^{5} + \left(18 a + 40\right)\cdot 59^{6} + \left(17 a + 3\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 5 }$ $=$ $26 a + 49 + \left(9 a + 29\right)\cdot 59 + \left(23 a + 23\right)\cdot 59^{2} + \left(35 a + 34\right)\cdot 59^{3} + \left(29 a + 48\right)\cdot 59^{4} + \left(54 a + 54\right)\cdot 59^{5} + \left(40 a + 53\right)\cdot 59^{6} + \left(41 a + 2\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 6 }$ $=$ $37 a + 11 + \left(54 a + 50\right)\cdot 59 + \left(20 a + 16\right)\cdot 59^{2} + \left(36 a + 51\right)\cdot 59^{3} + \left(36 a + 58\right)\cdot 59^{4} + \left(17 a + 38\right)\cdot 59^{5} + \left(23 a + 26\right)\cdot 59^{6} + \left(34 a + 53\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 7 }$ $=$ $3 + 50\cdot 59 + 7\cdot 59^{3} + 33\cdot 59^{4} + 4\cdot 59^{5} + 35\cdot 59^{6} + 22\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 8 }$ $=$ $26 a + 43 + \left(9 a + 45\right)\cdot 59 + \left(23 a + 21\right)\cdot 59^{2} + \left(35 a + 12\right)\cdot 59^{3} + \left(29 a + 16\right)\cdot 59^{4} + \left(54 a + 38\right)\cdot 59^{5} + \left(40 a + 18\right)\cdot 59^{6} + \left(41 a + 55\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$

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

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

### Character values on conjugacy classes

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