# Properties

 Label 2.2e4_3e2_17.24t22.1c2 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{4} \cdot 3^{2} \cdot 17$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $2448= 2^{4} \cdot 3^{2} \cdot 17$ Artin number field: Splitting field of $f= x^{8} + 12 x^{6} - 6 x^{4} - 48 x^{2} - 51$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.3_17.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $x^{2} + 58 x + 2$
Roots:
 $r_{ 1 }$ $=$ $49 a + 5 + \left(50 a + 58\right)\cdot 59 + \left(56 a + 55\right)\cdot 59^{2} + \left(6 a + 24\right)\cdot 59^{3} + \left(15 a + 25\right)\cdot 59^{4} + \left(6 a + 4\right)\cdot 59^{5} + \left(58 a + 33\right)\cdot 59^{6} + \left(42 a + 7\right)\cdot 59^{7} + \left(33 a + 34\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ $r_{ 2 }$ $=$ $22 a + 18 + \left(14 a + 21\right)\cdot 59 + \left(52 a + 51\right)\cdot 59^{2} + \left(7 a + 6\right)\cdot 59^{3} + \left(50 a + 29\right)\cdot 59^{4} + \left(7 a + 41\right)\cdot 59^{5} + \left(52 a + 56\right)\cdot 59^{6} + \left(6 a + 29\right)\cdot 59^{7} + \left(14 a + 36\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ $r_{ 3 }$ $=$ $22 a + 19 + \left(14 a + 45\right)\cdot 59 + \left(52 a + 28\right)\cdot 59^{2} + \left(7 a + 37\right)\cdot 59^{3} + \left(50 a + 46\right)\cdot 59^{4} + 7 a\cdot 59^{5} + \left(52 a + 17\right)\cdot 59^{6} + \left(6 a + 15\right)\cdot 59^{7} + \left(14 a + 15\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ $r_{ 4 }$ $=$ $23 + 19\cdot 59 + 49\cdot 59^{2} + 22\cdot 59^{3} + 30\cdot 59^{4} + 32\cdot 59^{5} + 10\cdot 59^{6} + 52\cdot 59^{7} + 23\cdot 59^{8} +O\left(59^{ 9 }\right)$ $r_{ 5 }$ $=$ $10 a + 54 + 8 a\cdot 59 + \left(2 a + 3\right)\cdot 59^{2} + \left(52 a + 34\right)\cdot 59^{3} + \left(43 a + 33\right)\cdot 59^{4} + \left(52 a + 54\right)\cdot 59^{5} + 25\cdot 59^{6} + \left(16 a + 51\right)\cdot 59^{7} + \left(25 a + 24\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ $r_{ 6 }$ $=$ $37 a + 41 + \left(44 a + 37\right)\cdot 59 + \left(6 a + 7\right)\cdot 59^{2} + \left(51 a + 52\right)\cdot 59^{3} + \left(8 a + 29\right)\cdot 59^{4} + \left(51 a + 17\right)\cdot 59^{5} + \left(6 a + 2\right)\cdot 59^{6} + \left(52 a + 29\right)\cdot 59^{7} + \left(44 a + 22\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ $r_{ 7 }$ $=$ $37 a + 40 + \left(44 a + 13\right)\cdot 59 + \left(6 a + 30\right)\cdot 59^{2} + \left(51 a + 21\right)\cdot 59^{3} + \left(8 a + 12\right)\cdot 59^{4} + \left(51 a + 58\right)\cdot 59^{5} + \left(6 a + 41\right)\cdot 59^{6} + \left(52 a + 43\right)\cdot 59^{7} + \left(44 a + 43\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ $r_{ 8 }$ $=$ $36 + 39\cdot 59 + 9\cdot 59^{2} + 36\cdot 59^{3} + 28\cdot 59^{4} + 26\cdot 59^{5} + 48\cdot 59^{6} + 6\cdot 59^{7} + 35\cdot 59^{8} +O\left(59^{ 9 }\right)$

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

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

### Character values on conjugacy classes

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