# Properties

 Label 2.2e3_43.24t22.1c1 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{3} \cdot 43$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $344= 2^{3} \cdot 43$ Artin number field: Splitting field of $f=x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 6 x^{3} - 2 x^{2} - 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.43.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$
Roots: \begin{aligned} r_{ 1 } &= 484966176 a - 1231961661 +O\left(37^{ 6 }\right) \\ r_{ 2 } &= 816564238 a + 106783280 +O\left(37^{ 6 }\right) \\ r_{ 3 } &= 930307922 a + 163145809 +O\left(37^{ 6 }\right) \\ r_{ 4 } &= -113556636 +O\left(37^{ 6 }\right) \\ r_{ 5 } &= 1282221011 +O\left(37^{ 6 }\right) \\ r_{ 6 } &= -816564238 a - 1182572484 +O\left(37^{ 6 }\right) \\ r_{ 7 } &= -930307922 a + 251701291 +O\left(37^{ 6 }\right) \\ r_{ 8 } &= -484966176 a + 724239394 +O\left(37^{ 6 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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