Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 19\cdot 59 + 22\cdot 59^{2} + 7\cdot 59^{3} + 59^{4} + 48\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 36\cdot 59 + 37\cdot 59^{2} + 58\cdot 59^{3} + 25\cdot 59^{4} + 39\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 30\cdot 59 + 29\cdot 59^{2} + 34\cdot 59^{3} + 45\cdot 59^{4} + 3\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 47\cdot 59 + 44\cdot 59^{2} + 26\cdot 59^{3} + 11\cdot 59^{4} + 54\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 11\cdot 59 + 14\cdot 59^{2} + 32\cdot 59^{3} + 47\cdot 59^{4} + 4\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 + 28\cdot 59 + 29\cdot 59^{2} + 24\cdot 59^{3} + 13\cdot 59^{4} + 55\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 52 + 22\cdot 59 + 21\cdot 59^{2} + 33\cdot 59^{4} + 19\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 39\cdot 59 + 36\cdot 59^{2} + 51\cdot 59^{3} + 57\cdot 59^{4} + 10\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,4)(2,3,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
