Basic invariants
Dimension: | $2$ |
Group: | $\textrm{GL(2,3)}$ |
Conductor: | \(2151\)\(\medspace = 3^{2} \cdot 239 \) |
Artin stem field: | Galois closure of 8.2.9952248951.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T22 |
Parity: | odd |
Determinant: | 1.239.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.2151.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 4x^{6} + 2x^{5} - 8x^{4} + 8x^{3} + 10x^{2} - 13x - 5 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 29 a + 24 + \left(38 a + 22\right)\cdot 43 + \left(33 a + 31\right)\cdot 43^{2} + \left(20 a + 41\right)\cdot 43^{3} + \left(3 a + 19\right)\cdot 43^{4} + \left(42 a + 22\right)\cdot 43^{5} + \left(38 a + 41\right)\cdot 43^{6} + \left(17 a + 38\right)\cdot 43^{7} +O(43^{8})\) |
$r_{ 2 }$ | $=$ | \( 9 + 34\cdot 43 + 23\cdot 43^{2} + 16\cdot 43^{3} + 16\cdot 43^{5} + 3\cdot 43^{6} + 12\cdot 43^{7} +O(43^{8})\) |
$r_{ 3 }$ | $=$ | \( 35 + 8\cdot 43 + 19\cdot 43^{2} + 26\cdot 43^{3} + 42\cdot 43^{4} + 26\cdot 43^{5} + 39\cdot 43^{6} + 30\cdot 43^{7} +O(43^{8})\) |
$r_{ 4 }$ | $=$ | \( 22 a + 11 + \left(18 a + 23\right)\cdot 43 + \left(26 a + 17\right)\cdot 43^{2} + \left(23 a + 1\right)\cdot 43^{3} + \left(7 a + 8\right)\cdot 43^{4} + \left(23 a + 35\right)\cdot 43^{5} + \left(11 a + 5\right)\cdot 43^{6} + \left(40 a + 7\right)\cdot 43^{7} +O(43^{8})\) |
$r_{ 5 }$ | $=$ | \( 21 a + 33 + \left(24 a + 19\right)\cdot 43 + \left(16 a + 25\right)\cdot 43^{2} + \left(19 a + 41\right)\cdot 43^{3} + \left(35 a + 34\right)\cdot 43^{4} + \left(19 a + 7\right)\cdot 43^{5} + \left(31 a + 37\right)\cdot 43^{6} + \left(2 a + 35\right)\cdot 43^{7} +O(43^{8})\) |
$r_{ 6 }$ | $=$ | \( 14 a + 10 + \left(4 a + 32\right)\cdot 43 + \left(9 a + 26\right)\cdot 43^{2} + \left(22 a + 28\right)\cdot 43^{3} + \left(39 a + 2\right)\cdot 43^{4} + 18\cdot 43^{5} + \left(4 a + 38\right)\cdot 43^{6} + \left(25 a + 17\right)\cdot 43^{7} +O(43^{8})\) |
$r_{ 7 }$ | $=$ | \( 29 a + 34 + \left(38 a + 10\right)\cdot 43 + \left(33 a + 16\right)\cdot 43^{2} + \left(20 a + 14\right)\cdot 43^{3} + \left(3 a + 40\right)\cdot 43^{4} + \left(42 a + 24\right)\cdot 43^{5} + \left(38 a + 4\right)\cdot 43^{6} + \left(17 a + 25\right)\cdot 43^{7} +O(43^{8})\) |
$r_{ 8 }$ | $=$ | \( 14 a + 20 + \left(4 a + 20\right)\cdot 43 + \left(9 a + 11\right)\cdot 43^{2} + \left(22 a + 1\right)\cdot 43^{3} + \left(39 a + 23\right)\cdot 43^{4} + 20\cdot 43^{5} + \left(4 a + 1\right)\cdot 43^{6} + \left(25 a + 4\right)\cdot 43^{7} +O(43^{8})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $-2$ |
$12$ | $2$ | $(1,8)(4,7)(5,6)$ | $0$ |
$8$ | $3$ | $(1,4,2)(3,8,5)$ | $-1$ |
$6$ | $4$ | $(1,6,8,7)(2,5,3,4)$ | $0$ |
$8$ | $6$ | $(1,3,4,8,2,5)(6,7)$ | $1$ |
$6$ | $8$ | $(1,7,2,5,8,6,3,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
$6$ | $8$ | $(1,6,2,4,8,7,3,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.