Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(2151\)\(\medspace = 3^{2} \cdot 239 \) |
| Artin number field: | Galois closure of 8.2.9952248951.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.2151.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 38 a + 20 + \left(42 a + 5\right)\cdot 43 + \left(16 a + 23\right)\cdot 43^{2} + \left(31 a + 8\right)\cdot 43^{3} + \left(42 a + 7\right)\cdot 43^{4} + \left(36 a + 3\right)\cdot 43^{5} + 16\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 + 21\cdot 43 + 3\cdot 43^{2} + 25\cdot 43^{3} + 42\cdot 43^{4} + 31\cdot 43^{5} + 17\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 8\cdot 43 + 5\cdot 43^{2} + 15\cdot 43^{3} + 7\cdot 43^{4} + 8\cdot 43^{5} + 4\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 a + 38 + \left(20 a + 36\right)\cdot 43 + \left(33 a + 39\right)\cdot 43^{2} + \left(18 a + 25\right)\cdot 43^{3} + \left(20 a + 35\right)\cdot 43^{4} + \left(9 a + 37\right)\cdot 43^{5} + \left(38 a + 25\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a + 27 + \left(a + 36\right)\cdot 43 + \left(8 a + 21\right)\cdot 43^{2} + \left(4 a + 33\right)\cdot 43^{3} + \left(20 a + 3\right)\cdot 43^{4} + \left(36 a + 25\right)\cdot 43^{5} + \left(6 a + 8\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 15 + 10\cdot 43 + \left(26 a + 40\right)\cdot 43^{2} + \left(11 a + 22\right)\cdot 43^{3} + 18\cdot 43^{4} + \left(6 a + 40\right)\cdot 43^{5} + \left(42 a + 22\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 a + 31 + \left(22 a + 21\right)\cdot 43 + \left(9 a + 9\right)\cdot 43^{2} + \left(24 a + 11\right)\cdot 43^{3} + \left(22 a + 37\right)\cdot 43^{4} + \left(33 a + 26\right)\cdot 43^{5} + \left(4 a + 11\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 36 a + 34 + \left(41 a + 30\right)\cdot 43 + \left(34 a + 28\right)\cdot 43^{2} + \left(38 a + 29\right)\cdot 43^{3} + \left(22 a + 19\right)\cdot 43^{4} + \left(6 a + 41\right)\cdot 43^{5} + \left(36 a + 21\right)\cdot 43^{6} +O(43^{7})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $-2$ | $-2$ |
| $12$ | $2$ | $(2,8)(3,4)(5,7)$ | $0$ | $0$ |
| $8$ | $3$ | $(1,2,8)(3,4,6)$ | $-1$ | $-1$ |
| $6$ | $4$ | $(1,4,6,8)(2,5,3,7)$ | $0$ | $0$ |
| $8$ | $6$ | $(1,5,3,6,7,2)(4,8)$ | $1$ | $1$ |
| $6$ | $8$ | $(1,3,5,4,6,2,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,5,8,6,3,7,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |