# Properties

 Label 2.751.24t22.1c1 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $751$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $751$ Artin number field: Splitting field of $f=x^{8} - x^{7} - 2 x^{6} + 4 x^{5} - 9 x^{4} + 17 x^{3} - 21 x^{2} + 7 x - 4$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.751.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots: \begin{aligned} r_{ 1 } &= 25797713 a - 38737840 +O\left(29^{ 6 }\right) \\ r_{ 2 } &= 16040346 a + 185971938 +O\left(29^{ 6 }\right) \\ r_{ 3 } &= 234519565 +O\left(29^{ 6 }\right) \\ r_{ 4 } &= -16040346 a - 198996366 +O\left(29^{ 6 }\right) \\ r_{ 5 } &= 109634464 a + 81623935 +O\left(29^{ 6 }\right) \\ r_{ 6 } &= -31011683 +O\left(29^{ 6 }\right) \\ r_{ 7 } &= -25797713 a - 63059631 +O\left(29^{ 6 }\right) \\ r_{ 8 } &= -109634464 a - 170309917 +O\left(29^{ 6 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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