Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 271 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 83 + 193\cdot 271 + 228\cdot 271^{2} + 5\cdot 271^{3} + 163\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 146 + 177\cdot 271 + 59\cdot 271^{2} + 74\cdot 271^{3} + 63\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 154 + 152\cdot 271 + 211\cdot 271^{2} + 188\cdot 271^{3} + 126\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 160 + 18\cdot 271 + 42\cdot 271^{2} + 2\cdot 271^{3} + 189\cdot 271^{4} +O\left(271^{ 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.
