Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 487 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 128 + 250\cdot 487 + 389\cdot 487^{2} + 431\cdot 487^{3} + 390\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 186 + 407\cdot 487 + 388\cdot 487^{2} + 261\cdot 487^{3} + 146\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 291 + 264\cdot 487 + 269\cdot 487^{2} + 279\cdot 487^{3} + 412\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 421 + 213\cdot 487 + 229\cdot 487^{2} + 153\cdot 487^{3} + 76\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 435 + 324\cdot 487 + 183\cdot 487^{2} + 334\cdot 487^{3} + 434\cdot 487^{4} +O\left(487^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
