Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 16\cdot 223 + 129\cdot 223^{2} + 177\cdot 223^{3} + 21\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 65 + 191\cdot 223 + 94\cdot 223^{2} + 77\cdot 223^{3} + 139\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 67 + 50\cdot 223 + 9\cdot 223^{2} + 156\cdot 223^{3} + 86\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 76 + 139\cdot 223 + 168\cdot 223^{2} + 69\cdot 223^{3} + 209\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 147 + 83\cdot 223 + 54\cdot 223^{2} + 153\cdot 223^{3} + 13\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 156 + 172\cdot 223 + 213\cdot 223^{2} + 66\cdot 223^{3} + 136\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 158 + 31\cdot 223 + 128\cdot 223^{2} + 145\cdot 223^{3} + 83\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 219 + 206\cdot 223 + 93\cdot 223^{2} + 45\cdot 223^{3} + 201\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
