Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 8\cdot 23 + 20\cdot 23^{2} + 8\cdot 23^{3} + 16\cdot 23^{4} + 17\cdot 23^{5} + 20\cdot 23^{6} + 15\cdot 23^{7} + 19\cdot 23^{8} + 6\cdot 23^{9} +O\left(23^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 19\cdot 23 + 16\cdot 23^{2} + 4\cdot 23^{3} + 16\cdot 23^{4} + 14\cdot 23^{5} + 19\cdot 23^{6} + 3\cdot 23^{7} + 20\cdot 23^{8} + 22\cdot 23^{9} +O\left(23^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 8\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 19\cdot 23^{4} + 5\cdot 23^{5} + 9\cdot 23^{6} + 15\cdot 23^{7} + 14\cdot 23^{8} + 17\cdot 23^{9} +O\left(23^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 11\cdot 23 + 21\cdot 23^{2} + 10\cdot 23^{3} + 6\cdot 23^{4} + 20\cdot 23^{5} + 17\cdot 23^{6} + 3\cdot 23^{7} + 14\cdot 23^{8} + 19\cdot 23^{9} +O\left(23^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 11\cdot 23 + 23^{2} + 12\cdot 23^{3} + 16\cdot 23^{4} + 2\cdot 23^{5} + 5\cdot 23^{6} + 19\cdot 23^{7} + 8\cdot 23^{8} + 3\cdot 23^{9} +O\left(23^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 + 14\cdot 23 + 5\cdot 23^{3} + 3\cdot 23^{4} + 17\cdot 23^{5} + 13\cdot 23^{6} + 7\cdot 23^{7} + 8\cdot 23^{8} + 5\cdot 23^{9} +O\left(23^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 14 + 3\cdot 23 + 6\cdot 23^{2} + 18\cdot 23^{3} + 6\cdot 23^{4} + 8\cdot 23^{5} + 3\cdot 23^{6} + 19\cdot 23^{7} + 2\cdot 23^{8} +O\left(23^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 20 + 14\cdot 23 + 2\cdot 23^{2} + 14\cdot 23^{3} + 6\cdot 23^{4} + 5\cdot 23^{5} + 2\cdot 23^{6} + 7\cdot 23^{7} + 3\cdot 23^{8} + 16\cdot 23^{9} +O\left(23^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,7,8,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
