Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 251 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 + 114\cdot 251^{2} + 222\cdot 251^{3} + 73\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 6\cdot 251 + 134\cdot 251^{2} + 68\cdot 251^{3} + 186\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 58 + 181\cdot 251 + 246\cdot 251^{2} + 82\cdot 251^{3} + 47\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 82 + 29\cdot 251 + 179\cdot 251^{2} + 198\cdot 251^{3} + 231\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 135 + 142\cdot 251 + 64\cdot 251^{2} + 167\cdot 251^{3} + 156\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 174 + 30\cdot 251 + 117\cdot 251^{2} + 189\cdot 251^{3} + 73\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 225 + 239\cdot 251 + 212\cdot 251^{2} + 77\cdot 251^{3} + 239\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 246 + 122\cdot 251 + 186\cdot 251^{2} + 247\cdot 251^{3} + 245\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,7,2,3)(4,5,8,6)$ |
| $(1,8,2,4)(3,6,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $4$ | $(1,8,2,4)(3,6,7,5)$ | $0$ |
| $2$ | $4$ | $(1,7,2,3)(4,5,8,6)$ | $0$ |
| $2$ | $4$ | $(1,6,2,5)(3,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
