Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 23\cdot 43 + 42\cdot 43^{2} + 2\cdot 43^{3} + 15\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 40\cdot 43 + 32\cdot 43^{2} + 21\cdot 43^{3} + 27\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 32\cdot 43 + 19\cdot 43^{2} + 31\cdot 43^{3} + 28\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 24\cdot 43 + 22\cdot 43^{2} + 14\cdot 43^{3} + 41\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 18\cdot 43 + 20\cdot 43^{2} + 28\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 10\cdot 43 + 23\cdot 43^{2} + 11\cdot 43^{3} + 14\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 38 + 2\cdot 43 + 10\cdot 43^{2} + 21\cdot 43^{3} + 15\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 19\cdot 43 + 40\cdot 43^{3} + 27\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
