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