# Properties

 Label 2.2e3_11e2.24t22.1 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{3} \cdot 11^{2}$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $968= 2^{3} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{8} + 22 x^{4} - 44 x^{3} - 44 x - 99$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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