Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 117\cdot 157 + 97\cdot 157^{2} + 42\cdot 157^{3} + 2\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 + 29\cdot 157 + 40\cdot 157^{2} + 75\cdot 157^{3} + 125\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 58 + 135\cdot 157 + 53\cdot 157^{2} + 6\cdot 157^{3} + 146\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 14\cdot 157 + 112\cdot 157^{2} + 3\cdot 157^{3} + 25\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 89 + 142\cdot 157 + 44\cdot 157^{2} + 153\cdot 157^{3} + 131\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 99 + 21\cdot 157 + 103\cdot 157^{2} + 150\cdot 157^{3} + 10\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 110 + 127\cdot 157 + 116\cdot 157^{2} + 81\cdot 157^{3} + 31\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 140 + 39\cdot 157 + 59\cdot 157^{2} + 114\cdot 157^{3} + 154\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
