Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42\cdot 43 + 23\cdot 43^{2} + 29\cdot 43^{3} + 38\cdot 43^{4} + 25\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 43 + 19\cdot 43^{2} + 13\cdot 43^{3} + 4\cdot 43^{4} + 17\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 21\cdot 43 + 20\cdot 43^{2} + 24\cdot 43^{3} + 8\cdot 43^{4} + 36\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 26\cdot 43 + 38\cdot 43^{2} + 11\cdot 43^{3} + 27\cdot 43^{4} + 16\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 3\cdot 43 + 40\cdot 43^{2} + 22\cdot 43^{3} + 31\cdot 43^{4} + 35\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 39\cdot 43 + 2\cdot 43^{2} + 20\cdot 43^{3} + 11\cdot 43^{4} + 7\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 16\cdot 43 + 4\cdot 43^{2} + 31\cdot 43^{3} + 15\cdot 43^{4} + 26\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 38 + 21\cdot 43 + 22\cdot 43^{2} + 18\cdot 43^{3} + 34\cdot 43^{4} + 6\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3,4,6)(2,8,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,4,6)(2,8,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
