Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 17\cdot 23 + 8\cdot 23^{2} + 22\cdot 23^{3} + 7\cdot 23^{4} + 4\cdot 23^{5} + 19\cdot 23^{6} + 11\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 12\cdot 23 + 11\cdot 23^{2} + 11\cdot 23^{4} + 14\cdot 23^{5} + 15\cdot 23^{6} + 16\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 2\cdot 23 + 16\cdot 23^{2} + 19\cdot 23^{3} + 21\cdot 23^{4} + 16\cdot 23^{5} + 3\cdot 23^{6} + 10\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 21\cdot 23 + 15\cdot 23^{2} + 13\cdot 23^{3} + 7\cdot 23^{4} + 5\cdot 23^{6} + 7\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 + 17\cdot 23 + 11\cdot 23^{2} + 13\cdot 23^{3} + 8\cdot 23^{4} + 8\cdot 23^{5} + 8\cdot 23^{6} + 20\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 12\cdot 23 + 14\cdot 23^{2} + 14\cdot 23^{3} + 11\cdot 23^{4} + 18\cdot 23^{5} + 4\cdot 23^{6} + 2\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 + 17\cdot 23 + 6\cdot 23^{2} + 18\cdot 23^{3} + 18\cdot 23^{4} + 22\cdot 23^{5} + 16\cdot 23^{6} + 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 21 + 13\cdot 23 + 6\cdot 23^{2} + 12\cdot 23^{3} + 4\cdot 23^{4} + 6\cdot 23^{5} + 18\cdot 23^{6} + 21\cdot 23^{7} +O\left(23^{ 8 }\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.
