Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 38\cdot 59 + 44\cdot 59^{2} + 52\cdot 59^{3} + 21\cdot 59^{4} + 40\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 45\cdot 59 + 7\cdot 59^{2} + 47\cdot 59^{3} + 21\cdot 59^{4} + 3\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 16\cdot 59 + 18\cdot 59^{3} + 3\cdot 59^{4} + 25\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 8\cdot 59 + 18\cdot 59^{2} + 57\cdot 59^{3} + 14\cdot 59^{4} + 48\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 15\cdot 59 + 40\cdot 59^{2} + 51\cdot 59^{3} + 14\cdot 59^{4} + 11\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 48\cdot 59 + 32\cdot 59^{2} + 54\cdot 59^{3} + 18\cdot 59^{4} + 41\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 36 + 6\cdot 59 + 45\cdot 59^{2} + 28\cdot 59^{3} + 12\cdot 59^{4} + 49\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 57\cdot 59 + 46\cdot 59^{2} + 43\cdot 59^{3} + 9\cdot 59^{4} + 17\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,5)(6,7)$ |
| $(1,3,5,6)(2,7,4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,5,6)(2,7,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
