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