Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 9\cdot 53 + 24\cdot 53^{2} + 16\cdot 53^{3} + 43\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 9\cdot 53 + 34\cdot 53^{2} + 48\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 + 38\cdot 53 + 16\cdot 53^{2} + 40\cdot 53^{3} + 16\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 47\cdot 53 + 30\cdot 53^{2} + 37\cdot 53^{4} +O\left(53^{ 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.
