# Properties

 Label 2.1557.24t22.b.b Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $3^{2} \cdot 173$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $1557= 3^{2} \cdot 173$ Artin number field: Splitting field of 8.2.11323667079.3 defined by $f= x^{8} - 6 x^{6} - 12 x^{5} - 12 x^{4} - 24 x^{3} - 24 x^{2} - 9 x - 6$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.519.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.2.4671.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$
Roots:
 $r_{ 1 }$ $=$ $1 + 15\cdot 23 + 3\cdot 23^{2} + 23^{3} + 23^{4} + 10\cdot 23^{5} + 3\cdot 23^{6} + 3\cdot 23^{7} +O\left(23^{ 8 }\right)$ $r_{ 2 }$ $=$ $12 a + 15 + \left(2 a + 1\right)\cdot 23 + 14 a\cdot 23^{2} + \left(18 a + 11\right)\cdot 23^{3} + \left(2 a + 5\right)\cdot 23^{4} + \left(12 a + 19\right)\cdot 23^{5} + \left(20 a + 1\right)\cdot 23^{6} + \left(2 a + 17\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ $r_{ 3 }$ $=$ $7 a + 1 + \left(15 a + 7\right)\cdot 23 + \left(14 a + 4\right)\cdot 23^{2} + 9 a\cdot 23^{4} + \left(2 a + 21\right)\cdot 23^{5} + \left(21 a + 2\right)\cdot 23^{6} + \left(3 a + 22\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ $r_{ 4 }$ $=$ $16 a + 15 + \left(7 a + 7\right)\cdot 23 + \left(8 a + 18\right)\cdot 23^{2} + \left(22 a + 9\right)\cdot 23^{3} + \left(13 a + 17\right)\cdot 23^{4} + \left(20 a + 16\right)\cdot 23^{5} + \left(a + 19\right)\cdot 23^{6} + \left(19 a + 8\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ $r_{ 5 }$ $=$ $18 + 8\cdot 23 + 14\cdot 23^{2} + 7\cdot 23^{3} + 19\cdot 23^{4} + 6\cdot 23^{5} + 16\cdot 23^{6} + 11\cdot 23^{7} +O\left(23^{ 8 }\right)$ $r_{ 6 }$ $=$ $20 a + 16 + \left(3 a + 11\right)\cdot 23 + 14\cdot 23^{2} + \left(3 a + 22\right)\cdot 23^{3} + \left(14 a + 3\right)\cdot 23^{4} + \left(22 a + 19\right)\cdot 23^{5} + \left(17 a + 1\right)\cdot 23^{6} + \left(5 a + 5\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ $r_{ 7 }$ $=$ $3 a + 10 + \left(19 a + 22\right)\cdot 23 + \left(22 a + 10\right)\cdot 23^{2} + \left(19 a + 5\right)\cdot 23^{3} + \left(8 a + 6\right)\cdot 23^{4} + 4\cdot 23^{5} + \left(5 a + 15\right)\cdot 23^{6} + \left(17 a + 21\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ $r_{ 8 }$ $=$ $11 a + 16 + \left(20 a + 17\right)\cdot 23 + \left(8 a + 2\right)\cdot 23^{2} + \left(4 a + 11\right)\cdot 23^{3} + \left(20 a + 15\right)\cdot 23^{4} + \left(10 a + 17\right)\cdot 23^{5} + \left(2 a + 7\right)\cdot 23^{6} + \left(20 a + 2\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$

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

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

### Character values on conjugacy classes

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