Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 281 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 228\cdot 281 + 22\cdot 281^{2} + 275\cdot 281^{3} + 3\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 275\cdot 281 + 18\cdot 281^{2} + 187\cdot 281^{3} + 19\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 + 93\cdot 281 + 223\cdot 281^{2} + 159\cdot 281^{3} + 71\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 257\cdot 281 + 80\cdot 281^{2} + 256\cdot 281^{3} + 219\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 200 + 238\cdot 281 + 87\cdot 281^{2} + 29\cdot 281^{3} + 38\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 214 + 4\cdot 281 + 84\cdot 281^{2} + 222\cdot 281^{3} + 53\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 258 + 71\cdot 281 + 93\cdot 281^{2} + 89\cdot 281^{3} + 3\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 265 + 235\cdot 281 + 231\cdot 281^{2} + 185\cdot 281^{3} + 151\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,6,8)(2,7,5,4)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,6,8)(2,7,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
