Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(2284\)\(\medspace = 2^{2} \cdot 571 \) |
| Artin number field: | Galois closure of 8.2.2978710576.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.2284.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 22 + \left(21 a + 14\right)\cdot 41 + \left(3 a + 5\right)\cdot 41^{2} + \left(2 a + 21\right)\cdot 41^{3} + \left(40 a + 31\right)\cdot 41^{4} + \left(7 a + 32\right)\cdot 41^{5} + \left(a + 5\right)\cdot 41^{6} + \left(28 a + 33\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 a + 20 + \left(19 a + 26\right)\cdot 41 + \left(37 a + 35\right)\cdot 41^{2} + \left(38 a + 19\right)\cdot 41^{3} + 9\cdot 41^{4} + \left(33 a + 8\right)\cdot 41^{5} + \left(39 a + 35\right)\cdot 41^{6} + \left(12 a + 7\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 37 + \left(21 a + 11\right)\cdot 41 + \left(3 a + 5\right)\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(40 a + 14\right)\cdot 41^{4} + \left(7 a + 24\right)\cdot 41^{5} + \left(a + 39\right)\cdot 41^{6} + \left(28 a + 6\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a + 19 + \left(26 a + 29\right)\cdot 41 + \left(23 a + 18\right)\cdot 41^{2} + \left(29 a + 8\right)\cdot 41^{3} + \left(12 a + 16\right)\cdot 41^{4} + \left(6 a + 17\right)\cdot 41^{5} + \left(25 a + 6\right)\cdot 41^{6} + \left(10 a + 17\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 32\cdot 41 + 18\cdot 41^{2} + 17\cdot 41^{3} + 8\cdot 41^{4} + 2\cdot 41^{5} + 28\cdot 41^{6} + 33\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 5 + \left(19 a + 29\right)\cdot 41 + \left(37 a + 35\right)\cdot 41^{2} + \left(38 a + 23\right)\cdot 41^{3} + 26\cdot 41^{4} + \left(33 a + 16\right)\cdot 41^{5} + \left(39 a + 1\right)\cdot 41^{6} + \left(12 a + 34\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 37 + 8\cdot 41 + 22\cdot 41^{2} + 23\cdot 41^{3} + 32\cdot 41^{4} + 38\cdot 41^{5} + 12\cdot 41^{6} + 7\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 26 a + 23 + \left(14 a + 11\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(11 a + 32\right)\cdot 41^{3} + \left(28 a + 24\right)\cdot 41^{4} + \left(34 a + 23\right)\cdot 41^{5} + \left(15 a + 34\right)\cdot 41^{6} + \left(30 a + 23\right)\cdot 41^{7} +O(41^{8})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $-2$ | $-2$ |
| $12$ | $2$ | $(1,2)(4,7)(5,8)$ | $0$ | $0$ |
| $8$ | $3$ | $(1,6,7)(2,3,5)$ | $-1$ | $-1$ |
| $6$ | $4$ | $(1,6,2,3)(4,5,8,7)$ | $0$ | $0$ |
| $8$ | $6$ | $(1,5,6,2,7,3)(4,8)$ | $1$ | $1$ |
| $6$ | $8$ | $(1,5,3,4,2,7,6,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,7,3,8,2,5,6,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |