Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 27\cdot 59 + 17\cdot 59^{2} + 23\cdot 59^{3} + 17\cdot 59^{4} + 18\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 22\cdot 59 + 20\cdot 59^{2} + 38\cdot 59^{3} + 21\cdot 59^{4} + 15\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 3\cdot 59 + 30\cdot 59^{2} + 19\cdot 59^{3} + 26\cdot 59^{4} + 5\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 17\cdot 59 + 57\cdot 59^{2} + 4\cdot 59^{3} + 38\cdot 59^{4} + 34\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 41\cdot 59 + 59^{2} + 54\cdot 59^{3} + 20\cdot 59^{4} + 24\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 55\cdot 59 + 28\cdot 59^{2} + 39\cdot 59^{3} + 32\cdot 59^{4} + 53\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 36\cdot 59 + 38\cdot 59^{2} + 20\cdot 59^{3} + 37\cdot 59^{4} + 43\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 31\cdot 59 + 41\cdot 59^{2} + 35\cdot 59^{3} + 41\cdot 59^{4} + 40\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,6)(3,8,4,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,8,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
