Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 23\cdot 43 + 32\cdot 43^{2} + 19\cdot 43^{3} + 32\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 10\cdot 43 + 18\cdot 43^{2} + 6\cdot 43^{3} + 27\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 7\cdot 43 + 8\cdot 43^{2} + 35\cdot 43^{3} + 2\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 40\cdot 43 + 34\cdot 43^{3} + 21\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 3\cdot 43 + 24\cdot 43^{2} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 25\cdot 43 + 27\cdot 43^{2} + 26\cdot 43^{3} + 22\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 + 16\cdot 43 + 26\cdot 43^{2} + 12\cdot 43^{3} + 40\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 34\cdot 43^{2} + 36\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,7,8)(2,4,6,5)$ |
| $(1,2)(3,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
