Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 359 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 220 + 318\cdot 359 + 244\cdot 359^{2} + 35\cdot 359^{3} + 280\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 273 + 111\cdot 359 + 218\cdot 359^{2} + 143\cdot 359^{3} + 188\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 292 + 124\cdot 359 + 2\cdot 359^{2} + 104\cdot 359^{3} + 72\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 293 + 162\cdot 359 + 252\cdot 359^{2} + 75\cdot 359^{3} + 177\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.
