Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 86 + 230\cdot 337 + 58\cdot 337^{2} + 3\cdot 337^{3} + 327\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 154 + 302\cdot 337 + 331\cdot 337^{2} + 155\cdot 337^{3} + 58\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 205 + 240\cdot 337 + 96\cdot 337^{2} + 168\cdot 337^{3} + 325\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 230 + 237\cdot 337 + 186\cdot 337^{2} + 9\cdot 337^{3} + 300\cdot 337^{4} +O\left(337^{ 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.
