Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 499 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 121 + 359\cdot 499 + 431\cdot 499^{2} + 56\cdot 499^{3} + 376\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 123 + 266\cdot 499 + 309\cdot 499^{2} + 381\cdot 499^{3} + 84\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 163 + 125\cdot 499 + 219\cdot 499^{2} + 146\cdot 499^{3} + 173\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 180 + 69\cdot 499 + 489\cdot 499^{2} + 150\cdot 499^{3} +O\left(499^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 412 + 177\cdot 499 + 47\cdot 499^{2} + 262\cdot 499^{3} + 363\cdot 499^{4} +O\left(499^{ 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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
