Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 293 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 134 + 23\cdot 293 + 118\cdot 293^{2} + 234\cdot 293^{3} + 53\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 148 + 12\cdot 293 + 22\cdot 293^{2} + 283\cdot 293^{3} + 171\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 154 + 150\cdot 293 + 141\cdot 293^{2} + 262\cdot 293^{3} + 195\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 218 + 7\cdot 293 + 255\cdot 293^{2} + 240\cdot 293^{3} + 260\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 227 + 98\cdot 293 + 49\cdot 293^{2} + 151\cdot 293^{3} + 196\cdot 293^{4} +O\left(293^{ 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.
