Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1557\)\(\medspace = 3^{2} \cdot 173 \) |
| Artin number field: | Galois closure of 8.2.11323667079.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.4671.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 5 + \left(15 a + 4\right)\cdot 23 + \left(22 a + 16\right)\cdot 23^{2} + \left(13 a + 18\right)\cdot 23^{3} + \left(16 a + 18\right)\cdot 23^{4} + \left(13 a + 11\right)\cdot 23^{5} + 11 a\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 14 + \left(3 a + 22\right)\cdot 23 + \left(6 a + 3\right)\cdot 23^{2} + \left(19 a + 20\right)\cdot 23^{3} + \left(9 a + 19\right)\cdot 23^{4} + \left(16 a + 22\right)\cdot 23^{5} + \left(6 a + 11\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 21 + \left(7 a + 3\right)\cdot 23 + \left(9 a + 1\right)\cdot 23^{3} + \left(6 a + 15\right)\cdot 23^{4} + \left(9 a + 22\right)\cdot 23^{5} + \left(11 a + 9\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 6 + \left(9 a + 3\right)\cdot 23 + \left(2 a + 7\right)\cdot 23^{2} + \left(21 a + 4\right)\cdot 23^{3} + \left(5 a + 3\right)\cdot 23^{4} + \left(6 a + 3\right)\cdot 23^{5} + \left(9 a + 10\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + 20 + \left(19 a + 2\right)\cdot 23 + \left(16 a + 13\right)\cdot 23^{2} + \left(3 a + 6\right)\cdot 23^{3} + \left(13 a + 20\right)\cdot 23^{4} + \left(6 a + 22\right)\cdot 23^{5} + \left(16 a + 8\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 + 18\cdot 23 + 14\cdot 23^{2} + 20\cdot 23^{3} + 14\cdot 23^{4} + 15\cdot 23^{5} + 14\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 19 + 16\cdot 23 + 10\cdot 23^{2} + 22\cdot 23^{3} + 5\cdot 23^{4} + 6\cdot 23^{5} + 13\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 a + 10 + \left(13 a + 19\right)\cdot 23 + \left(20 a + 2\right)\cdot 23^{2} + \left(a + 21\right)\cdot 23^{3} + \left(17 a + 16\right)\cdot 23^{4} + \left(16 a + 9\right)\cdot 23^{5} + \left(13 a + 22\right)\cdot 23^{6} +O(23^{7})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ | $-2$ |
| $12$ | $2$ | $(1,4)(2,8)(3,5)$ | $0$ | $0$ |
| $8$ | $3$ | $(2,8,6)(3,7,5)$ | $-1$ | $-1$ |
| $6$ | $4$ | $(1,5,4,2)(3,7,8,6)$ | $0$ | $0$ |
| $8$ | $6$ | $(1,3,5,4,8,2)(6,7)$ | $1$ | $1$ |
| $6$ | $8$ | $(1,3,7,2,4,8,6,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,8,7,5,4,3,6,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |