Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3175\)\(\medspace = 5^{2} \cdot 127 \) |
| Artin stem field: | Galois closure of 8.2.1280239375.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.127.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.3175.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 5x^{5} - 6x^{4} - 10x^{3} + 21x^{2} + 5x - 13 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 5 a + 10 + \left(13 a + 21\right)\cdot 23 + \left(13 a + 14\right)\cdot 23^{2} + \left(5 a + 2\right)\cdot 23^{3} + \left(4 a + 10\right)\cdot 23^{4} + \left(7 a + 6\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 11 + 17 a\cdot 23 + 17 a\cdot 23^{2} + \left(16 a + 13\right)\cdot 23^{3} + 21 a\cdot 23^{4} + \left(17 a + 21\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 22 + \left(19 a + 7\right)\cdot 23 + \left(a + 9\right)\cdot 23^{2} + \left(13 a + 16\right)\cdot 23^{3} + \left(19 a + 20\right)\cdot 23^{4} + \left(17 a + 22\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 20 + \left(9 a + 19\right)\cdot 23 + \left(9 a + 5\right)\cdot 23^{2} + 17 a\cdot 23^{3} + \left(18 a + 13\right)\cdot 23^{4} + \left(15 a + 16\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + 5 + \left(3 a + 20\right)\cdot 23 + \left(21 a + 16\right)\cdot 23^{2} + \left(9 a + 17\right)\cdot 23^{3} + 3 a\cdot 23^{4} + \left(5 a + 16\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 5 + \left(5 a + 16\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(6 a + 5\right)\cdot 23^{3} + \left(a + 4\right)\cdot 23^{4} + \left(5 a + 12\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 + 15\cdot 23 + 16\cdot 23^{2} + 19\cdot 23^{3} + 20\cdot 23^{4} + 5\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 14 + 13\cdot 23 + 10\cdot 23^{2} + 16\cdot 23^{3} + 21\cdot 23^{4} + 13\cdot 23^{5} +O(23^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $-2$ |
| $12$ | $2$ | $(1,3)(5,6)(7,8)$ | $0$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ |
| $6$ | $4$ | $(1,7,6,8)(2,5,4,3)$ | $0$ |
| $8$ | $6$ | $(1,4,3,6,2,5)(7,8)$ | $1$ |
| $6$ | $8$ | $(1,8,3,2,6,7,5,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,7,3,4,6,8,5,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.