Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 11\cdot 43^{2} + 21\cdot 43^{3} + 41\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 3\cdot 43 + 37\cdot 43^{2} + 7\cdot 43^{3} + 15\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 25\cdot 43 + 30\cdot 43^{2} + 19\cdot 43^{3} + 21\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 + 28\cdot 43 + 13\cdot 43^{2} + 6\cdot 43^{3} + 38\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 14\cdot 43 + 29\cdot 43^{2} + 36\cdot 43^{3} + 4\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 + 17\cdot 43 + 12\cdot 43^{2} + 23\cdot 43^{3} + 21\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 + 39\cdot 43 + 5\cdot 43^{2} + 35\cdot 43^{3} + 27\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 + 42\cdot 43 + 31\cdot 43^{2} + 21\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,4)(2,3,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
