Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 + 69\cdot 283 + 22\cdot 283^{2} + 2\cdot 283^{3} + 179\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 74 + 269\cdot 283 + 166\cdot 283^{2} + 55\cdot 283^{3} + 150\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 102 + 244\cdot 283 + 196\cdot 283^{2} + 186\cdot 283^{3} + 10\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 131 + 161\cdot 283 + 58\cdot 283^{2} + 240\cdot 283^{3} + 264\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 152 + 121\cdot 283 + 224\cdot 283^{2} + 42\cdot 283^{3} + 18\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 181 + 38\cdot 283 + 86\cdot 283^{2} + 96\cdot 283^{3} + 272\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 209 + 13\cdot 283 + 116\cdot 283^{2} + 227\cdot 283^{3} + 132\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 238 + 213\cdot 283 + 260\cdot 283^{2} + 280\cdot 283^{3} + 103\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.
