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