Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 48\cdot 61 + 21\cdot 61^{2} + 31\cdot 61^{3} + 3\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 48\cdot 61 + 41\cdot 61^{2} + 36\cdot 61^{3} + 30\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 18\cdot 61 + 19\cdot 61^{2} + 59\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 6\cdot 61 + 39\cdot 61^{2} + 55\cdot 61^{3} + 54\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
