Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 125\cdot 157 + 115\cdot 157^{2} + 59\cdot 157^{3} + 40\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 70\cdot 157 + 89\cdot 157^{2} + 40\cdot 157^{3} + 55\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 38\cdot 157 + 106\cdot 157^{2} + 133\cdot 157^{3} + 21\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 76\cdot 157 + 154\cdot 157^{2} + 76\cdot 157^{3} + 117\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 136 + 80\cdot 157 + 2\cdot 157^{2} + 80\cdot 157^{3} + 39\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 146 + 118\cdot 157 + 50\cdot 157^{2} + 23\cdot 157^{3} + 135\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 149 + 86\cdot 157 + 67\cdot 157^{2} + 116\cdot 157^{3} + 101\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 155 + 31\cdot 157 + 41\cdot 157^{2} + 97\cdot 157^{3} + 116\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,5)(4,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
