Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 10\cdot 43 + 41\cdot 43^{2} + 26\cdot 43^{3} + 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 40\cdot 43 + 24\cdot 43^{2} + 13\cdot 43^{3} + 21\cdot 43^{4} + 39\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 24\cdot 43 + 20\cdot 43^{2} + 12\cdot 43^{3} + 42\cdot 43^{4} + 18\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 24\cdot 43 + 37\cdot 43^{2} + 9\cdot 43^{3} + 13\cdot 43^{4} + 35\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 18\cdot 43 + 5\cdot 43^{2} + 33\cdot 43^{3} + 29\cdot 43^{4} + 7\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 18\cdot 43 + 22\cdot 43^{2} + 30\cdot 43^{3} + 24\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 2\cdot 43 + 18\cdot 43^{2} + 29\cdot 43^{3} + 21\cdot 43^{4} + 3\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 32\cdot 43 + 43^{2} + 16\cdot 43^{3} + 42\cdot 43^{4} + 41\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,3,7)(2,8,5,6)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,4,3,7)(2,8,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
