Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 240\cdot 337 + 231\cdot 337^{2} + 234\cdot 337^{3} + 190\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 172\cdot 337 + 42\cdot 337^{2} + 18\cdot 337^{3} + 286\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 180 + 52\cdot 337 + 192\cdot 337^{2} + 200\cdot 337^{3} + 336\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 203 + 332\cdot 337 + 260\cdot 337^{2} + 4\cdot 337^{3} + 51\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 261 + 213\cdot 337 + 283\cdot 337^{2} + 215\cdot 337^{3} + 146\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
