Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.2.139968000000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.10800.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 4x^{6} - 10x^{5} + 40x^{4} - 40x^{3} + 10x^{2} - 10x - 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 a + 21 + \left(4 a + 18\right)\cdot 41 + \left(40 a + 13\right)\cdot 41^{2} + \left(36 a + 8\right)\cdot 41^{3} + \left(20 a + 12\right)\cdot 41^{4} + \left(34 a + 40\right)\cdot 41^{5} + \left(40 a + 3\right)\cdot 41^{6} + \left(13 a + 13\right)\cdot 41^{7} + \left(9 a + 4\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 17 + \left(34 a + 12\right)\cdot 41 + \left(14 a + 17\right)\cdot 41^{2} + \left(34 a + 17\right)\cdot 41^{3} + \left(20 a + 11\right)\cdot 41^{4} + \left(27 a + 23\right)\cdot 41^{5} + \left(22 a + 13\right)\cdot 41^{6} + \left(24 a + 18\right)\cdot 41^{7} + \left(9 a + 11\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 12 + \left(6 a + 21\right)\cdot 41 + \left(26 a + 27\right)\cdot 41^{2} + \left(6 a + 23\right)\cdot 41^{3} + \left(20 a + 39\right)\cdot 41^{4} + \left(13 a + 2\right)\cdot 41^{5} + \left(18 a + 13\right)\cdot 41^{6} + \left(16 a + 28\right)\cdot 41^{7} + \left(31 a + 15\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 17 + \left(22 a + 9\right)\cdot 41 + \left(a + 29\right)\cdot 41^{2} + 17\cdot 41^{3} + \left(25 a + 5\right)\cdot 41^{4} + \left(34 a + 37\right)\cdot 41^{5} + \left(29 a + 38\right)\cdot 41^{6} + \left(19 a + 25\right)\cdot 41^{7} + \left(8 a + 15\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 + 25\cdot 41 + 28\cdot 41^{2} + 29\cdot 41^{3} + 4\cdot 41^{4} + 3\cdot 41^{5} + 30\cdot 41^{6} + 33\cdot 41^{7} + 8\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 20 + \left(36 a + 5\right)\cdot 41 + 6\cdot 41^{2} + \left(4 a + 38\right)\cdot 41^{3} + \left(20 a + 37\right)\cdot 41^{4} + \left(6 a + 40\right)\cdot 41^{5} + 9\cdot 41^{6} + \left(27 a + 14\right)\cdot 41^{7} + \left(31 a + 18\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 20 + 41 + 30\cdot 41^{2} + 12\cdot 41^{3} + 13\cdot 41^{4} + 23\cdot 41^{5} + 41^{6} + 16\cdot 41^{7} + 27\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 35 a + 35 + \left(18 a + 28\right)\cdot 41 + \left(39 a + 11\right)\cdot 41^{2} + \left(40 a + 16\right)\cdot 41^{3} + \left(15 a + 39\right)\cdot 41^{4} + \left(6 a + 33\right)\cdot 41^{5} + \left(11 a + 11\right)\cdot 41^{6} + \left(21 a + 14\right)\cdot 41^{7} + \left(32 a + 21\right)\cdot 41^{8} +O(41^{9})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $-2$ | |
| $12$ | $2$ | $(1,5)(4,7)(6,8)$ | $0$ | ✓ |
| $8$ | $3$ | $(1,8,3)(2,4,6)$ | $-1$ | |
| $6$ | $4$ | $(1,5,4,7)(2,8,3,6)$ | $0$ | |
| $8$ | $6$ | $(1,2,8,4,3,6)(5,7)$ | $1$ | |
| $6$ | $8$ | $(1,3,7,8,4,2,5,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ | |
| $6$ | $8$ | $(1,2,7,6,4,3,5,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |