Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 51\cdot 59 + 30\cdot 59^{2} + 39\cdot 59^{3} + 3\cdot 59^{4} + 8\cdot 59^{5} + 51\cdot 59^{6} + 21\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 21\cdot 59 + 15\cdot 59^{2} + 46\cdot 59^{3} + 27\cdot 59^{4} + 20\cdot 59^{5} + 46\cdot 59^{6} + 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 23\cdot 59 + 57\cdot 59^{2} + 47\cdot 59^{3} + 21\cdot 59^{4} + 14\cdot 59^{5} + 40\cdot 59^{6} + 20\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 21\cdot 59 + 14\cdot 59^{2} + 43\cdot 59^{3} + 5\cdot 59^{4} + 16\cdot 59^{5} + 39\cdot 59^{6} + 14\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
