Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 12\cdot 23 + 14\cdot 23^{2} + 11\cdot 23^{3} + 18\cdot 23^{4} + 16\cdot 23^{5} + 11\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 21\cdot 23^{2} + 8\cdot 23^{4} + 18\cdot 23^{5} + 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 4\cdot 23 + 14\cdot 23^{2} + 23^{3} + 13\cdot 23^{4} + 11\cdot 23^{5} + 22\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 16\cdot 23 + 16\cdot 23^{2} + 17\cdot 23^{3} + 5\cdot 23^{4} + 11\cdot 23^{5} + 18\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 + 6\cdot 23 + 6\cdot 23^{2} + 5\cdot 23^{3} + 17\cdot 23^{4} + 11\cdot 23^{5} + 4\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 18\cdot 23 + 8\cdot 23^{2} + 21\cdot 23^{3} + 9\cdot 23^{4} + 11\cdot 23^{5} +O\left(23^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 + 22\cdot 23 + 23^{2} + 22\cdot 23^{3} + 14\cdot 23^{4} + 4\cdot 23^{5} + 21\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 22 + 10\cdot 23 + 8\cdot 23^{2} + 11\cdot 23^{3} + 4\cdot 23^{4} + 6\cdot 23^{5} + 11\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
