Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 + 117\cdot 157 + 123\cdot 157^{2} + 58\cdot 157^{3} + 24\cdot 157^{4} + 21\cdot 157^{5} +O\left(157^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 + 13\cdot 157 + 62\cdot 157^{2} + 98\cdot 157^{3} + 43\cdot 157^{4} + 119\cdot 157^{5} +O\left(157^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 139\cdot 157 + 151\cdot 157^{2} + 90\cdot 157^{3} + 157^{4} + 67\cdot 157^{5} +O\left(157^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 + 45\cdot 157 + 89\cdot 157^{2} + 103\cdot 157^{3} + 134\cdot 157^{4} + 91\cdot 157^{5} +O\left(157^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 82 + 111\cdot 157 + 67\cdot 157^{2} + 53\cdot 157^{3} + 22\cdot 157^{4} + 65\cdot 157^{5} +O\left(157^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 108 + 17\cdot 157 + 5\cdot 157^{2} + 66\cdot 157^{3} + 155\cdot 157^{4} + 89\cdot 157^{5} +O\left(157^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 117 + 143\cdot 157 + 94\cdot 157^{2} + 58\cdot 157^{3} + 113\cdot 157^{4} + 37\cdot 157^{5} +O\left(157^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 124 + 39\cdot 157 + 33\cdot 157^{2} + 98\cdot 157^{3} + 132\cdot 157^{4} + 135\cdot 157^{5} +O\left(157^{ 6 }\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.
