Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 22\cdot 59 + 8\cdot 59^{2} + 20\cdot 59^{3} + 16\cdot 59^{4} + 39\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 54\cdot 59 + 44\cdot 59^{2} + 48\cdot 59^{3} + 14\cdot 59^{4} + 36\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 10\cdot 59 + 54\cdot 59^{2} + 59^{3} + 9\cdot 59^{4} + 40\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 27\cdot 59 + 48\cdot 59^{2} + 11\cdot 59^{3} + 40\cdot 59^{4} + 56\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 31\cdot 59 + 10\cdot 59^{2} + 47\cdot 59^{3} + 18\cdot 59^{4} + 2\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 + 48\cdot 59 + 4\cdot 59^{2} + 57\cdot 59^{3} + 49\cdot 59^{4} + 18\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 53 + 4\cdot 59 + 14\cdot 59^{2} + 10\cdot 59^{3} + 44\cdot 59^{4} + 22\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 36\cdot 59 + 50\cdot 59^{2} + 38\cdot 59^{3} + 42\cdot 59^{4} + 19\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,5)(4,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
