Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 51\cdot 157 + 96\cdot 157^{2} + 102\cdot 157^{3} + 45\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 122\cdot 157 + 151\cdot 157^{2} + 59\cdot 157^{3} + 69\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 119\cdot 157 + 88\cdot 157^{2} + 141\cdot 157^{3} + 48\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 24\cdot 157 + 89\cdot 157^{2} + 126\cdot 157^{3} + 109\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 100 + 132\cdot 157 + 67\cdot 157^{2} + 30\cdot 157^{3} + 47\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 131 + 37\cdot 157 + 68\cdot 157^{2} + 15\cdot 157^{3} + 108\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 146 + 34\cdot 157 + 5\cdot 157^{2} + 97\cdot 157^{3} + 87\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 151 + 105\cdot 157 + 60\cdot 157^{2} + 54\cdot 157^{3} + 111\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
