Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 271 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 + 15\cdot 271 + 235\cdot 271^{2} + 53\cdot 271^{3} + 198\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 93\cdot 271 + 6\cdot 271^{2} + 246\cdot 271^{3} + 198\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 118 + 45\cdot 271 + 66\cdot 271^{2} + 100\cdot 271^{3} + 25\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 122 + 206\cdot 271 + 56\cdot 271^{2} + 156\cdot 271^{3} + 55\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 149 + 64\cdot 271 + 214\cdot 271^{2} + 114\cdot 271^{3} + 215\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 153 + 225\cdot 271 + 204\cdot 271^{2} + 170\cdot 271^{3} + 245\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 204 + 177\cdot 271 + 264\cdot 271^{2} + 24\cdot 271^{3} + 72\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 227 + 255\cdot 271 + 35\cdot 271^{2} + 217\cdot 271^{3} + 72\cdot 271^{4} +O\left(271^{ 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.
