Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 5\cdot 23 + 15\cdot 23^{2} + 20\cdot 23^{3} + 18\cdot 23^{4} + 13\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 8\cdot 23 + 8\cdot 23^{2} + 14\cdot 23^{3} + 3\cdot 23^{4} + 2\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 11\cdot 23 + 12\cdot 23^{2} + 11\cdot 23^{3} + 11\cdot 23^{4} + 6\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 23 + 13\cdot 23^{2} + 11\cdot 23^{4} + 22\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 21\cdot 23 + 9\cdot 23^{2} + 22\cdot 23^{3} + 11\cdot 23^{4} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 11\cdot 23 + 10\cdot 23^{2} + 11\cdot 23^{3} + 11\cdot 23^{4} + 16\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 20 + 14\cdot 23 + 14\cdot 23^{2} + 8\cdot 23^{3} + 19\cdot 23^{4} + 20\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 21 + 17\cdot 23 + 7\cdot 23^{2} + 2\cdot 23^{3} + 4\cdot 23^{4} + 9\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,5,6)(2,4,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
