Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(2547\)\(\medspace = 3^{2} \cdot 283 \) |
| Artin stem field: | Galois closure of 8.2.1835880147.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.283.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.283.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 6x^{6} + 8x^{5} + 42x^{4} - 118x^{3} + 95x^{2} + 57x - 119 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4\cdot 17 + 8\cdot 17^{2} + 2\cdot 17^{3} + 9\cdot 17^{4} + 6\cdot 17^{5} + 8\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 7 + \left(10 a + 16\right)\cdot 17 + \left(6 a + 14\right)\cdot 17^{2} + \left(8 a + 2\right)\cdot 17^{3} + \left(11 a + 14\right)\cdot 17^{4} + 7\cdot 17^{5} + \left(15 a + 14\right)\cdot 17^{6} + \left(a + 1\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 15 + \left(5 a + 11\right)\cdot 17 + \left(a + 9\right)\cdot 17^{2} + 4 a\cdot 17^{3} + \left(5 a + 9\right)\cdot 17^{4} + \left(12 a + 3\right)\cdot 17^{5} + \left(13 a + 11\right)\cdot 17^{6} + \left(15 a + 7\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 + 12\cdot 17 + 7\cdot 17^{2} + 14\cdot 17^{3} + 2\cdot 17^{4} + 4\cdot 17^{5} + 5\cdot 17^{6} + 10\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 12 + \left(6 a + 13\right)\cdot 17 + \left(4 a + 6\right)\cdot 17^{2} + \left(8 a + 9\right)\cdot 17^{3} + \left(7 a + 11\right)\cdot 17^{4} + \left(14 a + 15\right)\cdot 17^{5} + \left(6 a + 9\right)\cdot 17^{6} + \left(5 a + 4\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 6 + \left(6 a + 11\right)\cdot 17 + \left(10 a + 10\right)\cdot 17^{2} + \left(8 a + 4\right)\cdot 17^{3} + 5 a\cdot 17^{4} + \left(16 a + 14\right)\cdot 17^{5} + \left(a + 11\right)\cdot 17^{6} + \left(15 a + 5\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 9 a + 3 + \left(10 a + 12\right)\cdot 17 + \left(12 a + 4\right)\cdot 17^{2} + \left(8 a + 13\right)\cdot 17^{3} + \left(9 a + 10\right)\cdot 17^{4} + \left(2 a + 5\right)\cdot 17^{5} + \left(10 a + 2\right)\cdot 17^{6} + \left(11 a + 3\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 2 a + 13 + \left(11 a + 2\right)\cdot 17 + \left(15 a + 5\right)\cdot 17^{2} + \left(12 a + 3\right)\cdot 17^{3} + \left(11 a + 10\right)\cdot 17^{4} + \left(4 a + 10\right)\cdot 17^{5} + \left(3 a + 12\right)\cdot 17^{6} + \left(a + 9\right)\cdot 17^{7} +O(17^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $-2$ |
| $12$ | $2$ | $(1,4)(2,3)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,7,6)(3,4,5)$ | $-1$ |
| $6$ | $4$ | $(1,7,4,5)(2,6,8,3)$ | $0$ |
| $8$ | $6$ | $(1,4)(2,6,7,8,3,5)$ | $1$ |
| $6$ | $8$ | $(1,6,5,2,4,3,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,3,5,8,4,6,7,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.