# Properties

 Label 2.1944.24t22.a Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{3} \cdot 3^{5}$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $1944= 2^{3} \cdot 3^{5}$ Artin number field: Splitting field of $f= x^{8} - 6 x^{4} + 4 x^{2} - 3$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Projective image: $S_4$ Projective field: Galois closure of 4.2.3888.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots:
 $r_{ 1 }$ $=$ $6 a + 14 + \left(17 a + 3\right)\cdot 29 + \left(27 a + 12\right)\cdot 29^{2} + \left(18 a + 24\right)\cdot 29^{3} + \left(25 a + 17\right)\cdot 29^{4} + \left(11 a + 26\right)\cdot 29^{5} + \left(16 a + 22\right)\cdot 29^{6} + \left(23 a + 21\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ $r_{ 2 }$ $=$ $4 + 23\cdot 29 + 17\cdot 29^{2} + 21\cdot 29^{3} + 27\cdot 29^{4} + 7\cdot 29^{5} + 22\cdot 29^{6} + 13\cdot 29^{7} +O\left(29^{ 8 }\right)$ $r_{ 3 }$ $=$ $a + 5 + \left(5 a + 10\right)\cdot 29 + \left(22 a + 15\right)\cdot 29^{2} + \left(3 a + 3\right)\cdot 29^{3} + \left(18 a + 17\right)\cdot 29^{4} + 15\cdot 29^{5} + \left(11 a + 19\right)\cdot 29^{6} + \left(22 a + 10\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ $r_{ 4 }$ $=$ $a + 19 + \left(5 a + 23\right)\cdot 29 + \left(22 a + 23\right)\cdot 29^{2} + \left(3 a + 28\right)\cdot 29^{3} + \left(18 a + 11\right)\cdot 29^{4} + 28\cdot 29^{5} + \left(11 a + 12\right)\cdot 29^{6} + \left(22 a + 4\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ $r_{ 5 }$ $=$ $23 a + 15 + \left(11 a + 25\right)\cdot 29 + \left(a + 16\right)\cdot 29^{2} + \left(10 a + 4\right)\cdot 29^{3} + \left(3 a + 11\right)\cdot 29^{4} + \left(17 a + 2\right)\cdot 29^{5} + \left(12 a + 6\right)\cdot 29^{6} + \left(5 a + 7\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ $r_{ 6 }$ $=$ $25 + 5\cdot 29 + 11\cdot 29^{2} + 7\cdot 29^{3} + 29^{4} + 21\cdot 29^{5} + 6\cdot 29^{6} + 15\cdot 29^{7} +O\left(29^{ 8 }\right)$ $r_{ 7 }$ $=$ $28 a + 24 + \left(23 a + 18\right)\cdot 29 + \left(6 a + 13\right)\cdot 29^{2} + \left(25 a + 25\right)\cdot 29^{3} + \left(10 a + 11\right)\cdot 29^{4} + \left(28 a + 13\right)\cdot 29^{5} + \left(17 a + 9\right)\cdot 29^{6} + \left(6 a + 18\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ $r_{ 8 }$ $=$ $28 a + 10 + \left(23 a + 5\right)\cdot 29 + \left(6 a + 5\right)\cdot 29^{2} + 25 a\cdot 29^{3} + \left(10 a + 17\right)\cdot 29^{4} + 28 a\cdot 29^{5} + \left(17 a + 16\right)\cdot 29^{6} + \left(6 a + 24\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$

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

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

### Character values on conjugacy classes

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