Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 13\cdot 157 + 37\cdot 157^{2} + 141\cdot 157^{3} + 102\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 5\cdot 157 + 60\cdot 157^{2} + 54\cdot 157^{3} + 54\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 7\cdot 157 + 146\cdot 157^{2} + 109\cdot 157^{3} + 23\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 137\cdot 157 + 138\cdot 157^{2} + 2\cdot 157^{3} + 108\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 + 47\cdot 157 + 109\cdot 157^{2} + 41\cdot 157^{3} + 126\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 134 + 108\cdot 157 + 134\cdot 157^{2} + 46\cdot 157^{3} + 72\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 138 + 121\cdot 157 + 76\cdot 157^{2} + 2\cdot 157^{3} + 56\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 153 + 29\cdot 157 + 82\cdot 157^{2} + 71\cdot 157^{3} + 84\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,8)(3,7,4,5)$ |
| $(1,6)(2,8)(3,4)(5,7)$ |
| $(1,4,8,7,6,3,2,5)$ |
| $(1,5,6,7)(2,4,8,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $-2$ |
| $4$ | $2$ | $(1,2)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,8,6,2)(3,5,4,7)$ | $0$ |
| $4$ | $4$ | $(1,7,6,5)(2,3,8,4)$ | $0$ |
| $2$ | $8$ | $(1,4,8,7,6,3,2,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,8,5,6,4,2,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
