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 8.2.5598720000.2 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} + 4x^{5} + 4x^{4} - 2x^{3} - 32x^{2} + 16x - 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 8 + \left(2 a + 21\right)\cdot 41 + \left(13 a + 13\right)\cdot 41^{2} + \left(a + 14\right)\cdot 41^{3} + \left(11 a + 40\right)\cdot 41^{4} + \left(5 a + 25\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 15 + \left(36 a + 7\right)\cdot 41 + \left(24 a + 33\right)\cdot 41^{2} + 11 a\cdot 41^{3} + \left(33 a + 27\right)\cdot 41^{4} + \left(34 a + 19\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 10 + \left(30 a + 38\right)\cdot 41 + \left(18 a + 35\right)\cdot 41^{2} + \left(13 a + 29\right)\cdot 41^{3} + \left(5 a + 38\right)\cdot 41^{4} + \left(34 a + 18\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 10 + \left(38 a + 1\right)\cdot 41 + \left(27 a + 9\right)\cdot 41^{2} + \left(39 a + 5\right)\cdot 41^{3} + \left(29 a + 31\right)\cdot 41^{4} + \left(35 a + 30\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 a + 39 + \left(4 a + 25\right)\cdot 41 + \left(16 a + 30\right)\cdot 41^{2} + \left(29 a + 10\right)\cdot 41^{3} + \left(7 a + 33\right)\cdot 41^{4} + \left(6 a + 8\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 + 14\cdot 41 + 18\cdot 41^{2} + 38\cdot 41^{3} + 31\cdot 41^{4} + 36\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 31 + 17\cdot 41 + 2\cdot 41^{2} + 13\cdot 41^{3} + 2\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 32 a + 37 + \left(10 a + 37\right)\cdot 41 + \left(22 a + 20\right)\cdot 41^{2} + \left(27 a + 10\right)\cdot 41^{3} + 35 a\cdot 41^{4} + \left(6 a + 34\right)\cdot 41^{5} +O(41^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $-2$ | |
| $12$ | $2$ | $(2,5)(4,7)(6,8)$ | $0$ | ✓ |
| $8$ | $3$ | $(1,6,5)(2,3,7)$ | $-1$ | |
| $6$ | $4$ | $(1,6,3,7)(2,4,5,8)$ | $0$ | |
| $8$ | $6$ | $(1,2,6,3,5,7)(4,8)$ | $1$ | |
| $6$ | $8$ | $(1,6,2,8,3,7,5,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ | |
| $6$ | $8$ | $(1,7,2,4,3,6,5,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |