Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| Artin stem field: | Galois closure of 8.2.9172942848.6 |
| 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 4.2.6912.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 8x^{6} + 18x^{4} - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 a + 27 + \left(24 a + 7\right)\cdot 29 + \left(22 a + 13\right)\cdot 29^{2} + \left(27 a + 14\right)\cdot 29^{3} + \left(17 a + 12\right)\cdot 29^{4} + \left(9 a + 28\right)\cdot 29^{5} + \left(15 a + 9\right)\cdot 29^{6} + \left(10 a + 10\right)\cdot 29^{7} + \left(20 a + 12\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 a + 7 + \left(28 a + 14\right)\cdot 29 + \left(a + 19\right)\cdot 29^{2} + \left(23 a + 12\right)\cdot 29^{3} + \left(9 a + 15\right)\cdot 29^{4} + \left(23 a + 27\right)\cdot 29^{5} + \left(22 a + 17\right)\cdot 29^{6} + \left(8 a + 17\right)\cdot 29^{7} + 16 a\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 19 + \left(28 a + 5\right)\cdot 29 + \left(a + 28\right)\cdot 29^{2} + \left(23 a + 18\right)\cdot 29^{3} + \left(9 a + 16\right)\cdot 29^{4} + \left(23 a + 10\right)\cdot 29^{5} + \left(22 a + 7\right)\cdot 29^{6} + \left(8 a + 19\right)\cdot 29^{7} + \left(16 a + 13\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 6\cdot 29^{2} + 28\cdot 29^{3} + 17\cdot 29^{4} + 8\cdot 29^{5} + 8\cdot 29^{6} + 10\cdot 29^{7} + 24\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 2 + \left(4 a + 21\right)\cdot 29 + \left(6 a + 15\right)\cdot 29^{2} + \left(a + 14\right)\cdot 29^{3} + \left(11 a + 16\right)\cdot 29^{4} + 19 a\cdot 29^{5} + \left(13 a + 19\right)\cdot 29^{6} + \left(18 a + 18\right)\cdot 29^{7} + \left(8 a + 16\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 22 + 14\cdot 29 + \left(27 a + 9\right)\cdot 29^{2} + \left(5 a + 16\right)\cdot 29^{3} + \left(19 a + 13\right)\cdot 29^{4} + \left(5 a + 1\right)\cdot 29^{5} + \left(6 a + 11\right)\cdot 29^{6} + \left(20 a + 11\right)\cdot 29^{7} + \left(12 a + 28\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 11 a + 10 + 23\cdot 29 + 27 a\cdot 29^{2} + \left(5 a + 10\right)\cdot 29^{3} + \left(19 a + 12\right)\cdot 29^{4} + \left(5 a + 18\right)\cdot 29^{5} + \left(6 a + 21\right)\cdot 29^{6} + \left(20 a + 9\right)\cdot 29^{7} + \left(12 a + 15\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 5 + 28\cdot 29 + 22\cdot 29^{2} + 11\cdot 29^{4} + 20\cdot 29^{5} + 20\cdot 29^{6} + 18\cdot 29^{7} + 4\cdot 29^{8} +O(29^{9})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ | |
| $12$ | $2$ | $(1,5)(2,8)(4,6)$ | $0$ | ✓ |
| $8$ | $3$ | $(1,4,2)(5,8,6)$ | $-1$ | |
| $6$ | $4$ | $(1,6,5,2)(3,4,7,8)$ | $0$ | |
| $8$ | $6$ | $(1,7,8,5,3,4)(2,6)$ | $1$ | |
| $6$ | $8$ | $(1,2,3,4,5,6,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | |
| $6$ | $8$ | $(1,6,3,8,5,2,7,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |