Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3528\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 8.2.263473523712.2 |
| 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 4.2.21168.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 4x^{6} + 34x^{5} + 58x^{4} + 10x^{3} - 62x^{2} - 56x - 14 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 10 + \left(7 a + 13\right)\cdot 17 + \left(8 a + 11\right)\cdot 17^{2} + \left(9 a + 9\right)\cdot 17^{3} + \left(12 a + 4\right)\cdot 17^{4} + \left(10 a + 3\right)\cdot 17^{5} + 7 a\cdot 17^{6} + \left(7 a + 5\right)\cdot 17^{7} + \left(15 a + 10\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 11 + \left(16 a + 1\right)\cdot 17 + \left(12 a + 14\right)\cdot 17^{2} + \left(12 a + 1\right)\cdot 17^{3} + \left(3 a + 16\right)\cdot 17^{4} + \left(7 a + 13\right)\cdot 17^{5} + \left(8 a + 4\right)\cdot 17^{6} + \left(4 a + 13\right)\cdot 17^{7} + \left(7 a + 6\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 13 + 9 a\cdot 17 + \left(8 a + 13\right)\cdot 17^{2} + \left(7 a + 10\right)\cdot 17^{3} + \left(4 a + 7\right)\cdot 17^{4} + \left(6 a + 1\right)\cdot 17^{5} + \left(9 a + 14\right)\cdot 17^{6} + \left(9 a + 4\right)\cdot 17^{7} + \left(a + 1\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + \left(10 a + 14\right)\cdot 17 + \left(7 a + 4\right)\cdot 17^{2} + \left(15 a + 15\right)\cdot 17^{3} + \left(a + 3\right)\cdot 17^{4} + \left(14 a + 9\right)\cdot 17^{5} + \left(10 a + 11\right)\cdot 17^{6} + \left(16 a + 8\right)\cdot 17^{7} + \left(6 a + 13\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 8 + \left(6 a + 16\right)\cdot 17 + \left(9 a + 1\right)\cdot 17^{2} + \left(a + 6\right)\cdot 17^{3} + \left(15 a + 7\right)\cdot 17^{4} + \left(2 a + 4\right)\cdot 17^{5} + \left(6 a + 8\right)\cdot 17^{6} + 14\cdot 17^{7} + \left(10 a + 3\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 + 8\cdot 17 + 5\cdot 17^{2} + 7\cdot 17^{3} + 7\cdot 17^{4} + 10\cdot 17^{5} + 3\cdot 17^{6} + 16\cdot 17^{7} + 3\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 6 + 2\cdot 17 + 6\cdot 17^{2} + 15\cdot 17^{3} + 13\cdot 17^{4} + 7\cdot 17^{5} + 2\cdot 17^{6} + 13\cdot 17^{7} + 17^{8} +O(17^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 10 a + 1 + 11\cdot 17 + \left(4 a + 10\right)\cdot 17^{2} + \left(4 a + 1\right)\cdot 17^{3} + \left(13 a + 7\right)\cdot 17^{4} + 9 a\cdot 17^{5} + \left(8 a + 6\right)\cdot 17^{6} + \left(12 a + 9\right)\cdot 17^{7} + \left(9 a + 9\right)\cdot 17^{8} +O(17^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-2$ |
| $12$ | $2$ | $(2,3)(4,8)(6,7)$ | $0$ |
| $8$ | $3$ | $(2,7,4)(3,8,6)$ | $-1$ |
| $6$ | $4$ | $(1,4,5,3)(2,7,8,6)$ | $0$ |
| $8$ | $6$ | $(1,4,7,5,3,6)(2,8)$ | $1$ |
| $6$ | $8$ | $(1,4,6,8,5,3,7,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,3,6,2,5,4,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.