Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 293 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 113 + 142\cdot 293 + 111\cdot 293^{2} + 147\cdot 293^{3} + 115\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 123 + 166\cdot 293 + 253\cdot 293^{2} + 163\cdot 293^{3} + 140\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 166 + 235\cdot 293 + 53\cdot 293^{2} + 119\cdot 293^{3} + 48\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 185 + 41\cdot 293 + 167\cdot 293^{2} + 155\cdot 293^{3} + 281\cdot 293^{4} +O\left(293^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
