Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 15\cdot 59 + 22\cdot 59^{2} + 48\cdot 59^{3} + 33\cdot 59^{4} + 26\cdot 59^{5} + 23\cdot 59^{6} + 52\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 49\cdot 59 + 10\cdot 59^{2} + 24\cdot 59^{3} + 47\cdot 59^{4} + 39\cdot 59^{5} + 12\cdot 59^{6} + 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 42\cdot 59 + 31\cdot 59^{2} + 8\cdot 59^{3} + 22\cdot 59^{4} + 52\cdot 59^{5} + 7\cdot 59^{6} + 24\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 + 10\cdot 59 + 53\cdot 59^{2} + 36\cdot 59^{3} + 14\cdot 59^{4} + 58\cdot 59^{5} + 14\cdot 59^{6} + 40\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,3)(2,4)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
