Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3528\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 8.2.5377010688.3 |
| 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 4.2.21168.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} - 2x^{5} + 4x^{4} + 22x^{3} + 10x^{2} - 26x - 17 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a + 18 + \left(15 a + 10\right)\cdot 23 + \left(10 a + 2\right)\cdot 23^{2} + \left(6 a + 13\right)\cdot 23^{3} + \left(8 a + 15\right)\cdot 23^{4} + \left(a + 4\right)\cdot 23^{5} + \left(3 a + 15\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 16\cdot 23 + 9\cdot 23^{2} + 6\cdot 23^{3} + 19\cdot 23^{4} + 23^{5} + 3\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 17 + \left(10 a + 5\right)\cdot 23 + \left(4 a + 17\right)\cdot 23^{2} + \left(17 a + 20\right)\cdot 23^{3} + \left(11 a + 11\right)\cdot 23^{4} + 10 a\cdot 23^{5} + \left(12 a + 3\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 5 + \left(14 a + 16\right)\cdot 23 + \left(11 a + 22\right)\cdot 23^{2} + \left(16 a + 5\right)\cdot 23^{3} + 19\cdot 23^{4} + \left(6 a + 11\right)\cdot 23^{5} + \left(3 a + 10\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 14 + \left(7 a + 21\right)\cdot 23 + \left(12 a + 7\right)\cdot 23^{2} + \left(16 a + 15\right)\cdot 23^{3} + \left(14 a + 2\right)\cdot 23^{4} + \left(21 a + 22\right)\cdot 23^{5} + \left(19 a + 19\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 10 + \left(12 a + 18\right)\cdot 23 + \left(18 a + 15\right)\cdot 23^{2} + \left(5 a + 4\right)\cdot 23^{3} + \left(11 a + 18\right)\cdot 23^{4} + \left(12 a + 9\right)\cdot 23^{5} + \left(10 a + 17\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 19 + 18\cdot 23 + 7\cdot 23^{2} + 21\cdot 23^{3} + 18\cdot 23^{5} + 11\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a + 14 + \left(8 a + 6\right)\cdot 23 + \left(11 a + 8\right)\cdot 23^{2} + \left(6 a + 4\right)\cdot 23^{3} + \left(22 a + 4\right)\cdot 23^{4} + 16 a\cdot 23^{5} + \left(19 a + 11\right)\cdot 23^{6} +O(23^{7})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $12$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $8$ | $3$ | $(1,6,7)(2,8,3)$ | $-1$ |
| $6$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $8$ | $6$ | $(1,2,6,8,7,3)(4,5)$ | $1$ |
| $6$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.