Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 281 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 89\cdot 281 + 89\cdot 281^{2} + 236\cdot 281^{3} + 166\cdot 281^{4} + 53\cdot 281^{5} + 94\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 73 + 126\cdot 281 + 197\cdot 281^{2} + 16\cdot 281^{3} + 110\cdot 281^{4} + 242\cdot 281^{5} + 64\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 121 + 107\cdot 281 + 164\cdot 281^{2} + 212\cdot 281^{3} + 100\cdot 281^{4} + 131\cdot 281^{5} + 20\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 134 + 108\cdot 281 + 8\cdot 281^{2} + 249\cdot 281^{3} + 13\cdot 281^{4} + 120\cdot 281^{5} + 162\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 147 + 172\cdot 281 + 272\cdot 281^{2} + 31\cdot 281^{3} + 267\cdot 281^{4} + 160\cdot 281^{5} + 118\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 160 + 173\cdot 281 + 116\cdot 281^{2} + 68\cdot 281^{3} + 180\cdot 281^{4} + 149\cdot 281^{5} + 260\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 208 + 154\cdot 281 + 83\cdot 281^{2} + 264\cdot 281^{3} + 170\cdot 281^{4} + 38\cdot 281^{5} + 216\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 278 + 191\cdot 281 + 191\cdot 281^{2} + 44\cdot 281^{3} + 114\cdot 281^{4} + 227\cdot 281^{5} + 186\cdot 281^{6} +O\left(281^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,2,5,6,8,7,4,3)$ |
| $(1,5,8,4)(2,6,7,3)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $0$ |
| $2$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $0$ |
| $2$ | $8$ | $(1,2,4,3,8,7,5,6)$ | $0$ |
| $2$ | $8$ | $(1,3,5,2,8,6,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
