Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Artin number field: | Galois closure of 8.2.181398528.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.3888.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 14 + \left(17 a + 3\right)\cdot 29 + \left(27 a + 12\right)\cdot 29^{2} + \left(18 a + 24\right)\cdot 29^{3} + \left(25 a + 17\right)\cdot 29^{4} + \left(11 a + 26\right)\cdot 29^{5} + \left(16 a + 22\right)\cdot 29^{6} + \left(23 a + 21\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 23\cdot 29 + 17\cdot 29^{2} + 21\cdot 29^{3} + 27\cdot 29^{4} + 7\cdot 29^{5} + 22\cdot 29^{6} + 13\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 5 + \left(5 a + 10\right)\cdot 29 + \left(22 a + 15\right)\cdot 29^{2} + \left(3 a + 3\right)\cdot 29^{3} + \left(18 a + 17\right)\cdot 29^{4} + 15\cdot 29^{5} + \left(11 a + 19\right)\cdot 29^{6} + \left(22 a + 10\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 19 + \left(5 a + 23\right)\cdot 29 + \left(22 a + 23\right)\cdot 29^{2} + \left(3 a + 28\right)\cdot 29^{3} + \left(18 a + 11\right)\cdot 29^{4} + 28\cdot 29^{5} + \left(11 a + 12\right)\cdot 29^{6} + \left(22 a + 4\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a + 15 + \left(11 a + 25\right)\cdot 29 + \left(a + 16\right)\cdot 29^{2} + \left(10 a + 4\right)\cdot 29^{3} + \left(3 a + 11\right)\cdot 29^{4} + \left(17 a + 2\right)\cdot 29^{5} + \left(12 a + 6\right)\cdot 29^{6} + \left(5 a + 7\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 + 5\cdot 29 + 11\cdot 29^{2} + 7\cdot 29^{3} + 29^{4} + 21\cdot 29^{5} + 6\cdot 29^{6} + 15\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a + 24 + \left(23 a + 18\right)\cdot 29 + \left(6 a + 13\right)\cdot 29^{2} + \left(25 a + 25\right)\cdot 29^{3} + \left(10 a + 11\right)\cdot 29^{4} + \left(28 a + 13\right)\cdot 29^{5} + \left(17 a + 9\right)\cdot 29^{6} + \left(6 a + 18\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 28 a + 10 + \left(23 a + 5\right)\cdot 29 + \left(6 a + 5\right)\cdot 29^{2} + 25 a\cdot 29^{3} + \left(10 a + 17\right)\cdot 29^{4} + 28 a\cdot 29^{5} + \left(17 a + 16\right)\cdot 29^{6} + \left(6 a + 24\right)\cdot 29^{7} +O(29^{8})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,4)(6,8)$ | $0$ | $0$ |
| $8$ | $3$ | $(2,4,7)(3,6,8)$ | $-1$ | $-1$ |
| $6$ | $4$ | $(1,8,5,4)(2,3,6,7)$ | $0$ | $0$ |
| $8$ | $6$ | $(1,5)(2,3,4,6,7,8)$ | $1$ | $1$ |
| $6$ | $8$ | $(1,6,7,4,5,2,3,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,7,8,5,6,3,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |