Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 359 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 92 + 114\cdot 359 + 330\cdot 359^{2} + 41\cdot 359^{3} + 294\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 145 + 128\cdot 359 + 341\cdot 359^{2} + 213\cdot 359^{3} + 110\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 152 + 154\cdot 359 + 96\cdot 359^{2} + 110\cdot 359^{3} + 56\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 329 + 320\cdot 359 + 308\cdot 359^{2} + 351\cdot 359^{3} + 256\cdot 359^{4} +O\left(359^{ 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.
