Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 499 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 60 + 473\cdot 499 + 371\cdot 499^{2} + 218\cdot 499^{3} + 133\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 249 + 388\cdot 499 + 127\cdot 499^{2} + 194\cdot 499^{3} + 137\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 300 + 105\cdot 499 + 388\cdot 499^{2} + 356\cdot 499^{3} + 143\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 402 + 461\cdot 499 + 494\cdot 499^{2} + 69\cdot 499^{3} + 22\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 487 + 67\cdot 499 + 114\cdot 499^{2} + 158\cdot 499^{3} + 62\cdot 499^{4} +O\left(499^{ 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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
