Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 509 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 + 224\cdot 509 + 396\cdot 509^{2} + 154\cdot 509^{3} + 48\cdot 509^{4} +O\left(509^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 173 + 8\cdot 509 + 7\cdot 509^{2} + 238\cdot 509^{3} + 113\cdot 509^{4} +O\left(509^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 205 + 327\cdot 509 + 399\cdot 509^{2} + 129\cdot 509^{3} + 291\cdot 509^{4} +O\left(509^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 242 + 186\cdot 509 + 473\cdot 509^{2} + 170\cdot 509^{3} + 104\cdot 509^{4} +O\left(509^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 352 + 271\cdot 509 + 250\cdot 509^{2} + 324\cdot 509^{3} + 460\cdot 509^{4} +O\left(509^{ 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.
