Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 5\cdot 157 + 128\cdot 157^{2} + 124\cdot 157^{3} + 60\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 120\cdot 157 + 100\cdot 157^{2} + 63\cdot 157^{3} + 28\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 59\cdot 157 + 102\cdot 157^{2} + 77\cdot 157^{3} + 69\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 96 + 129\cdot 157 + 139\cdot 157^{2} + 47\cdot 157^{3} + 155\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
