Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35 + 259\cdot 283 + 28\cdot 283^{2} + 53\cdot 283^{3} + 182\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 + 176\cdot 283 + 173\cdot 283^{2} + 106\cdot 283^{3} + 153\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 71 + 181\cdot 283 + 34\cdot 283^{2} + 216\cdot 283^{3} + 68\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 100 + 98\cdot 283 + 179\cdot 283^{2} + 269\cdot 283^{3} + 39\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 183 + 184\cdot 283 + 103\cdot 283^{2} + 13\cdot 283^{3} + 243\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 212 + 101\cdot 283 + 248\cdot 283^{2} + 66\cdot 283^{3} + 214\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 219 + 106\cdot 283 + 109\cdot 283^{2} + 176\cdot 283^{3} + 129\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 248 + 23\cdot 283 + 254\cdot 283^{2} + 229\cdot 283^{3} + 100\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,3,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
