Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 503 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 249\cdot 503 + 298\cdot 503^{2} + 57\cdot 503^{3} + 191\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 + 215\cdot 503 + 470\cdot 503^{2} + 361\cdot 503^{3} + 60\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 295 + 274\cdot 503 + 186\cdot 503^{2} + 344\cdot 503^{3} + 258\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 297 + 118\cdot 503 + 197\cdot 503^{2} + 380\cdot 503^{3} + 84\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 333 + 148\cdot 503 + 356\cdot 503^{2} + 364\cdot 503^{3} + 410\cdot 503^{4} +O\left(503^{ 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.
