Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 313 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 166 + 201\cdot 313 + 6\cdot 313^{2} + 25\cdot 313^{3} + 242\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 196 + 298\cdot 313 + 173\cdot 313^{2} + 281\cdot 313^{3} + 231\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 269 + 292\cdot 313 + 37\cdot 313^{2} + 307\cdot 313^{3} + 247\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 309 + 145\cdot 313 + 94\cdot 313^{2} + 12\cdot 313^{3} + 217\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
