Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 10\cdot 59 + 38\cdot 59^{2} + 35\cdot 59^{3} + 4\cdot 59^{4} + 47\cdot 59^{5} + 33\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 35\cdot 59 + 38\cdot 59^{2} + 42\cdot 59^{3} + 59^{4} + 57\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 43\cdot 59 + 46\cdot 59^{2} + 9\cdot 59^{3} + 32\cdot 59^{4} + 4\cdot 59^{5} + 48\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 29\cdot 59 + 53\cdot 59^{2} + 29\cdot 59^{3} + 20\cdot 59^{4} + 7\cdot 59^{5} + 38\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,3)$ | $0$ |
| $2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
