Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 150\cdot 157 + 138\cdot 157^{2} + 115\cdot 157^{3} + 153\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 9\cdot 157 + 38\cdot 157^{2} + 97\cdot 157^{3} + 115\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 54\cdot 157 + 64\cdot 157^{2} + 147\cdot 157^{3} + 13\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 126\cdot 157 + 2\cdot 157^{2} + 33\cdot 157^{3} + 93\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 120\cdot 157 + 110\cdot 157^{2} + 156\cdot 157^{3} + 25\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 50\cdot 157 + 7\cdot 157^{2} + 37\cdot 157^{3} + 86\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 + 11\cdot 157 + 24\cdot 157^{2} + 89\cdot 157^{3} + 88\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 140 + 105\cdot 157 + 84\cdot 157^{2} + 108\cdot 157^{3} + 50\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,3,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
