Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(2432\)\(\medspace = 2^{7} \cdot 19 \) |
| Artin stem field: | Galois closure of 8.2.28768731136.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.4864.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{4} + 16x^{2} - 76 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 + 15\cdot 59 + 54\cdot 59^{2} + 19\cdot 59^{3} + 16\cdot 59^{4} + 12\cdot 59^{5} + 33\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 50 a + 34 + \left(40 a + 4\right)\cdot 59 + \left(50 a + 54\right)\cdot 59^{2} + \left(28 a + 10\right)\cdot 59^{3} + \left(34 a + 56\right)\cdot 59^{4} + \left(58 a + 46\right)\cdot 59^{5} + \left(44 a + 6\right)\cdot 59^{6} + \left(36 a + 4\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 14 + \left(21 a + 42\right)\cdot 59 + \left(23 a + 1\right)\cdot 59^{2} + \left(10 a + 11\right)\cdot 59^{3} + \left(15 a + 42\right)\cdot 59^{4} + \left(16 a + 1\right)\cdot 59^{5} + \left(14 a + 20\right)\cdot 59^{6} + \left(44 a + 37\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 36 + \left(21 a + 4\right)\cdot 59 + \left(23 a + 55\right)\cdot 59^{2} + \left(10 a + 1\right)\cdot 59^{3} + \left(15 a + 12\right)\cdot 59^{4} + \left(16 a + 56\right)\cdot 59^{5} + \left(14 a + 40\right)\cdot 59^{6} + \left(44 a + 50\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 43\cdot 59 + 4\cdot 59^{2} + 39\cdot 59^{3} + 42\cdot 59^{4} + 46\cdot 59^{5} + 58\cdot 59^{6} + 25\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 25 + \left(18 a + 54\right)\cdot 59 + \left(8 a + 4\right)\cdot 59^{2} + \left(30 a + 48\right)\cdot 59^{3} + \left(24 a + 2\right)\cdot 59^{4} + 12\cdot 59^{5} + \left(14 a + 52\right)\cdot 59^{6} + \left(22 a + 54\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 50 a + 45 + \left(37 a + 16\right)\cdot 59 + \left(35 a + 57\right)\cdot 59^{2} + \left(48 a + 47\right)\cdot 59^{3} + \left(43 a + 16\right)\cdot 59^{4} + \left(42 a + 57\right)\cdot 59^{5} + \left(44 a + 38\right)\cdot 59^{6} + \left(14 a + 21\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 50 a + 23 + \left(37 a + 54\right)\cdot 59 + \left(35 a + 3\right)\cdot 59^{2} + \left(48 a + 57\right)\cdot 59^{3} + \left(43 a + 46\right)\cdot 59^{4} + \left(42 a + 2\right)\cdot 59^{5} + \left(44 a + 18\right)\cdot 59^{6} + \left(14 a + 8\right)\cdot 59^{7} +O(59^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,4)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,8,3)(4,7,5)$ | $-1$ |
| $6$ | $4$ | $(1,7,5,3)(2,4,6,8)$ | $0$ |
| $8$ | $6$ | $(1,7,8,5,3,4)(2,6)$ | $1$ |
| $6$ | $8$ | $(1,8,7,2,5,4,3,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,4,7,6,5,8,3,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.