# Properties

 Label 2.2151.24t22.a.a Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $2151$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$2151$$$$\medspace = 3^{2} \cdot 239$$ Artin number field: Galois closure of 8.2.9952248951.1 Galois orbit size: $2$ Smallest permutation container: 24T22 Parity: odd Determinant: 1.239.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.2.2151.1

## Defining polynomial

 $f(x)$ $=$ $x^{8} - 3 x^{7} - 8 x^{6} + 21 x^{5} + 21 x^{4} - 33 x^{3} - 14 x^{2} + 39 x - 8$.

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{2} + 42 x + 3$

Roots:
 $r_{ 1 }$ $=$ $38 a + 20 + \left(42 a + 5\right)\cdot 43 + \left(16 a + 23\right)\cdot 43^{2} + \left(31 a + 8\right)\cdot 43^{3} + \left(42 a + 7\right)\cdot 43^{4} + \left(36 a + 3\right)\cdot 43^{5} + 16\cdot 43^{6} +O\left(43^{ 7 }\right)$ $r_{ 2 }$ $=$ $41 + 21\cdot 43 + 3\cdot 43^{2} + 25\cdot 43^{3} + 42\cdot 43^{4} + 31\cdot 43^{5} + 17\cdot 43^{6} +O\left(43^{ 7 }\right)$ $r_{ 3 }$ $=$ $12 + 8\cdot 43 + 5\cdot 43^{2} + 15\cdot 43^{3} + 7\cdot 43^{4} + 8\cdot 43^{5} + 4\cdot 43^{6} +O\left(43^{ 7 }\right)$ $r_{ 4 }$ $=$ $36 a + 38 + \left(20 a + 36\right)\cdot 43 + \left(33 a + 39\right)\cdot 43^{2} + \left(18 a + 25\right)\cdot 43^{3} + \left(20 a + 35\right)\cdot 43^{4} + \left(9 a + 37\right)\cdot 43^{5} + \left(38 a + 25\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ $r_{ 5 }$ $=$ $7 a + 27 + \left(a + 36\right)\cdot 43 + \left(8 a + 21\right)\cdot 43^{2} + \left(4 a + 33\right)\cdot 43^{3} + \left(20 a + 3\right)\cdot 43^{4} + \left(36 a + 25\right)\cdot 43^{5} + \left(6 a + 8\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ $r_{ 6 }$ $=$ $5 a + 15 + 10\cdot 43 + \left(26 a + 40\right)\cdot 43^{2} + \left(11 a + 22\right)\cdot 43^{3} + 18\cdot 43^{4} + \left(6 a + 40\right)\cdot 43^{5} + \left(42 a + 22\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ $r_{ 7 }$ $=$ $7 a + 31 + \left(22 a + 21\right)\cdot 43 + \left(9 a + 9\right)\cdot 43^{2} + \left(24 a + 11\right)\cdot 43^{3} + \left(22 a + 37\right)\cdot 43^{4} + \left(33 a + 26\right)\cdot 43^{5} + \left(4 a + 11\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ $r_{ 8 }$ $=$ $36 a + 34 + \left(41 a + 30\right)\cdot 43 + \left(34 a + 28\right)\cdot 43^{2} + \left(38 a + 29\right)\cdot 43^{3} + \left(22 a + 19\right)\cdot 43^{4} + \left(6 a + 41\right)\cdot 43^{5} + \left(36 a + 21\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,6)(2,3)(4,8)(5,7)$ $-2$ $12$ $2$ $(2,8)(3,4)(5,7)$ $0$ $8$ $3$ $(1,2,8)(3,4,6)$ $-1$ $6$ $4$ $(1,4,6,8)(2,5,3,7)$ $0$ $8$ $6$ $(1,5,3,6,7,2)(4,8)$ $1$ $6$ $8$ $(1,3,5,4,6,2,7,8)$ $-\zeta_{8}^{3} - \zeta_{8}$ $6$ $8$ $(1,2,5,8,6,3,7,4)$ $\zeta_{8}^{3} + \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.