Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 58\cdot 59 + 18\cdot 59^{2} + 24\cdot 59^{3} + 19\cdot 59^{4} + 45\cdot 59^{5} + 28\cdot 59^{6} + 42\cdot 59^{7} + 46\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 33\cdot 59 + 44\cdot 59^{2} + 57\cdot 59^{3} + 14\cdot 59^{4} + 52\cdot 59^{5} + 13\cdot 59^{6} + 54\cdot 59^{7} + 15\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 2\cdot 59 + 36\cdot 59^{2} + 41\cdot 59^{3} + 17\cdot 59^{4} + 31\cdot 59^{5} + 31\cdot 59^{6} + 28\cdot 59^{7} + 14\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 5\cdot 59 + 53\cdot 59^{2} + 6\cdot 59^{3} + 31\cdot 59^{4} + 20\cdot 59^{5} + 45\cdot 59^{6} + 24\cdot 59^{7} + 20\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 53\cdot 59 + 5\cdot 59^{2} + 52\cdot 59^{3} + 27\cdot 59^{4} + 38\cdot 59^{5} + 13\cdot 59^{6} + 34\cdot 59^{7} + 38\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 + 56\cdot 59 + 22\cdot 59^{2} + 17\cdot 59^{3} + 41\cdot 59^{4} + 27\cdot 59^{5} + 27\cdot 59^{6} + 30\cdot 59^{7} + 44\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 43 + 25\cdot 59 + 14\cdot 59^{2} + 59^{3} + 44\cdot 59^{4} + 6\cdot 59^{5} + 45\cdot 59^{6} + 4\cdot 59^{7} + 43\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 40\cdot 59^{2} + 34\cdot 59^{3} + 39\cdot 59^{4} + 13\cdot 59^{5} + 30\cdot 59^{6} + 16\cdot 59^{7} + 12\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,4,6,8,2,5,3)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,8)(4,5)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ |
| $2$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
| $2$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $0$ |
| $2$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
