Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 + 67\cdot 283 + 157\cdot 283^{2} + 9\cdot 283^{3} + 61\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 154 + 11\cdot 283 + 124\cdot 283^{2} + 196\cdot 283^{3} + 198\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 180 + 118\cdot 283 + 256\cdot 283^{2} + 131\cdot 283^{3} + 99\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 198 + 254\cdot 283 + 20\cdot 283^{2} + 86\cdot 283^{3} + 137\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 261 + 113\cdot 283 + 7\cdot 283^{2} + 142\cdot 283^{3} + 69\cdot 283^{4} +O\left(283^{ 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 |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
