Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3\cdot 23 + 5\cdot 23^{2} + 23^{3} + 7\cdot 23^{4} + 17\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 14\cdot 23 + 16\cdot 23^{2} + 23^{3} + 12\cdot 23^{4} + 21\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 12\cdot 23 + 9\cdot 23^{2} + 6\cdot 23^{3} + 8\cdot 23^{4} + 13\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 22\cdot 23 + 18\cdot 23^{2} + 19\cdot 23^{3} + 14\cdot 23^{4} + 8\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 + 13\cdot 23 + 4\cdot 23^{2} + 11\cdot 23^{3} + 23^{4} + 19\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 + 20\cdot 23 + 18\cdot 23^{3} + 10\cdot 23^{4} + 2\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 + 6\cdot 23 + 5\cdot 23^{2} + 12\cdot 23^{4} + 21\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 21 + 22\cdot 23 + 7\cdot 23^{2} + 10\cdot 23^{3} + 2\cdot 23^{4} + 11\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,7,6)(2,8,4,5)$ |
| $(1,7)(2,4)(3,6)(5,8)$ |
| $(1,7)(2,8)(4,5)$ |
| $(1,5,3,2,7,8,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,4)(3,6)(5,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,8)(4,5)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,6)(2,8,4,5)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,4,7,2)(3,8,6,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,3,2,7,8,6,4)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,3,4,7,5,6,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
