Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1107\)\(\medspace = 3^{3} \cdot 41 \) |
| Artin stem field: | Galois closure of 8.2.150730227.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.123.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1107.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 2x^{6} - 2x^{5} - 3x^{4} - 7x^{3} - 11x^{2} - 6x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 13 + \left(12 a + 19\right)\cdot 23 + \left(19 a + 2\right)\cdot 23^{2} + \left(19 a + 15\right)\cdot 23^{3} + \left(13 a + 6\right)\cdot 23^{4} + \left(16 a + 9\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 11 + 10 a\cdot 23 + \left(3 a + 6\right)\cdot 23^{2} + \left(3 a + 12\right)\cdot 23^{3} + \left(9 a + 14\right)\cdot 23^{4} + \left(6 a + 5\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 17 + 11\cdot 23 + \left(22 a + 17\right)\cdot 23^{2} + \left(14 a + 1\right)\cdot 23^{3} + \left(14 a + 13\right)\cdot 23^{4} + \left(5 a + 8\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 17\cdot 23 + 4\cdot 23^{4} + 11\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 9 + \left(22 a + 17\right)\cdot 23 + 14\cdot 23^{2} + \left(8 a + 9\right)\cdot 23^{3} + \left(8 a + 4\right)\cdot 23^{4} + \left(17 a + 5\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 + 20\cdot 23 + 15\cdot 23^{2} + 22\cdot 23^{3} + 9\cdot 23^{4} + 16\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 6 a + \left(11 a + 17\right)\cdot 23 + \left(11 a + 22\right)\cdot 23^{2} + \left(12 a + 19\right)\cdot 23^{3} + \left(19 a + 17\right)\cdot 23^{4} + \left(2 a + 1\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 17 a + 12 + \left(11 a + 10\right)\cdot 23 + \left(11 a + 11\right)\cdot 23^{2} + \left(10 a + 10\right)\cdot 23^{3} + \left(3 a + 21\right)\cdot 23^{4} + \left(20 a + 10\right)\cdot 23^{5} +O(23^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $12$ | $2$ | $(2,3)(4,6)(5,8)$ | $0$ |
| $8$ | $3$ | $(1,5,8)(2,7,3)$ | $-1$ |
| $6$ | $4$ | $(1,2,7,8)(3,6,5,4)$ | $0$ |
| $8$ | $6$ | $(1,2,5,7,8,3)(4,6)$ | $1$ |
| $6$ | $8$ | $(1,3,4,2,7,5,6,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,5,4,8,7,3,6,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.