Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 90 + 201\cdot 283 + 210\cdot 283^{2} + 64\cdot 283^{3} + 258\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 147 + 107\cdot 283 + 153\cdot 283^{2} + 171\cdot 283^{3} + 222\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 151 + 145\cdot 283 + 191\cdot 283^{2} + 143\cdot 283^{3} + 80\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 180 + 111\cdot 283 + 10\cdot 283^{2} + 186\cdot 283^{3} + 4\cdot 283^{4} +O\left(283^{ 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.
