# Properties

 Label 2.643.24t22.1c2 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $643$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $643$ Artin number field: Splitting field of $f=x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.643.2t1.1c1

## 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 } &= 369330905 a + 345500228 +O\left(37^{ 6 }\right) \\ r_{ 2 } &= -369330905 a + 986212408 +O\left(37^{ 6 }\right) \\ r_{ 3 } &= 591780527 a + 201913555 +O\left(37^{ 6 }\right) \\ r_{ 4 } &= -150761675 a + 608472593 +O\left(37^{ 6 }\right) \\ r_{ 5 } &= 1007590881 +O\left(37^{ 6 }\right) \\ r_{ 6 } &= -591780527 a + 1198967436 +O\left(37^{ 6 }\right) \\ r_{ 7 } &= 330640669 +O\left(37^{ 6 }\right) \\ r_{ 8 } &= 150761675 a + 452155050 +O\left(37^{ 6 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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