# Properties

 Label 2.643.24t22.a Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $643$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$643$$ Artin number field: Galois closure of 8.2.265847707.1 Galois orbit size: $2$ Smallest permutation container: 24T22 Parity: odd Projective image: $S_4$ Projective field: Galois closure of 4.2.643.1

## 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:
 $r_{ 1 }$ $=$ $13 a + \left(19 a + 6\right)\cdot 37 + \left(14 a + 34\right)\cdot 37^{2} + \left(2 a + 12\right)\cdot 37^{3} + \left(12 a + 36\right)\cdot 37^{4} + \left(5 a + 4\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ $r_{ 2 }$ $=$ $24 a + 15 + \left(17 a + 33\right)\cdot 37 + \left(22 a + 35\right)\cdot 37^{2} + \left(34 a + 7\right)\cdot 37^{3} + \left(24 a + 8\right)\cdot 37^{4} + \left(31 a + 14\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ $r_{ 3 }$ $=$ $11 a + 4 + \left(4 a + 30\right)\cdot 37 + \left(a + 7\right)\cdot 37^{2} + \left(28 a + 27\right)\cdot 37^{3} + \left(19 a + 33\right)\cdot 37^{4} + \left(8 a + 2\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ $r_{ 4 }$ $=$ $5 a + 8 + 22 a\cdot 37 + \left(23 a + 21\right)\cdot 37^{2} + \left(20 a + 24\right)\cdot 37^{3} + \left(30 a + 28\right)\cdot 37^{4} + \left(34 a + 8\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ $r_{ 5 }$ $=$ $36 + 37^{2} + 23\cdot 37^{3} + 19\cdot 37^{4} + 14\cdot 37^{5} +O\left(37^{ 6 }\right)$ $r_{ 6 }$ $=$ $26 a + 11 + \left(32 a + 36\right)\cdot 37 + \left(35 a + 7\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(17 a + 10\right)\cdot 37^{4} + \left(28 a + 17\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ $r_{ 7 }$ $=$ $11 + 31\cdot 37 + 20\cdot 37^{2} + 15\cdot 37^{3} + 28\cdot 37^{4} + 4\cdot 37^{5} +O\left(37^{ 6 }\right)$ $r_{ 8 }$ $=$ $32 a + 28 + \left(14 a + 9\right)\cdot 37 + \left(13 a + 19\right)\cdot 37^{2} + \left(16 a + 9\right)\cdot 37^{3} + \left(6 a + 19\right)\cdot 37^{4} + \left(2 a + 6\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,2)(3,4)(5,7)(6,8)$ $-2$ $-2$ $12$ $2$ $(1,2)(5,8)(6,7)$ $0$ $0$ $8$ $3$ $(1,8,3)(2,6,4)$ $-1$ $-1$ $6$ $4$ $(1,7,2,5)(3,6,4,8)$ $0$ $0$ $8$ $6$ $(1,4,8,2,3,6)(5,7)$ $1$ $1$ $6$ $8$ $(1,5,3,6,2,7,4,8)$ $-\zeta_{8}^{3} - \zeta_{8}$ $\zeta_{8}^{3} + \zeta_{8}$ $6$ $8$ $(1,7,3,8,2,5,4,6)$ $\zeta_{8}^{3} + \zeta_{8}$ $-\zeta_{8}^{3} - \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.