# Properties

 Label 2.688.24t22.a.b Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{4} \cdot 43$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $688= 2^{4} \cdot 43$ Artin number field: Splitting field of 8.2.325660672.2 defined by $f= x^{8} - 2 x^{7} + 4 x^{6} - 12 x^{5} + 16 x^{4} - 18 x^{3} + 10 x^{2} - 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.43.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.2.688.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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