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