Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 6\cdot 43 + 41\cdot 43^{2} + 6\cdot 43^{3} + 17\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 38\cdot 43 + 13\cdot 43^{2} + 6\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 13\cdot 43 + 14\cdot 43^{2} + 24\cdot 43^{3} + 3\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 17\cdot 43 + 3\cdot 43^{2} + 2\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 33\cdot 43 + 17\cdot 43^{2} + 25\cdot 43^{3} + 12\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 33\cdot 43 + 38\cdot 43^{2} + 26\cdot 43^{3} + 28\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 40\cdot 43 + 31\cdot 43^{2} + 26\cdot 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 30\cdot 43 + 10\cdot 43^{2} + 10\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,8)(5,7)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,4,5,6)(2,3,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
