Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 11\cdot 59 + 54\cdot 59^{2} + 4\cdot 59^{3} + 46\cdot 59^{4} + 36\cdot 59^{5} + 22\cdot 59^{6} + 8\cdot 59^{7} + 39\cdot 59^{8} + 49\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 7\cdot 59 + 42\cdot 59^{2} + 22\cdot 59^{3} + 3\cdot 59^{4} + 44\cdot 59^{5} + 11\cdot 59^{6} + 24\cdot 59^{7} + 11\cdot 59^{8} + 28\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 10\cdot 59^{2} + 51\cdot 59^{3} + 27\cdot 59^{4} + 15\cdot 59^{5} + 56\cdot 59^{6} + 34\cdot 59^{7} + 40\cdot 59^{8} + 3\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 55\cdot 59 + 43\cdot 59^{2} + 7\cdot 59^{3} + 8\cdot 59^{4} + 26\cdot 59^{5} + 55\cdot 59^{6} + 27\cdot 59^{7} + 45\cdot 59^{8} + 30\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 3\cdot 59 + 15\cdot 59^{2} + 51\cdot 59^{3} + 50\cdot 59^{4} + 32\cdot 59^{5} + 3\cdot 59^{6} + 31\cdot 59^{7} + 13\cdot 59^{8} + 28\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 58\cdot 59 + 48\cdot 59^{2} + 7\cdot 59^{3} + 31\cdot 59^{4} + 43\cdot 59^{5} + 2\cdot 59^{6} + 24\cdot 59^{7} + 18\cdot 59^{8} + 55\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 46 + 51\cdot 59 + 16\cdot 59^{2} + 36\cdot 59^{3} + 55\cdot 59^{4} + 14\cdot 59^{5} + 47\cdot 59^{6} + 34\cdot 59^{7} + 47\cdot 59^{8} + 30\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 47\cdot 59 + 4\cdot 59^{2} + 54\cdot 59^{3} + 12\cdot 59^{4} + 22\cdot 59^{5} + 36\cdot 59^{6} + 50\cdot 59^{7} + 19\cdot 59^{8} + 9\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(2,7)(4,5)$ |
| $(1,5,6,2,8,4,3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,6,2,8,4,3,7)$ | $0$ |
| $2$ | $8$ | $(1,2,3,5,8,7,6,4)$ | $0$ |
| $2$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $0$ |
| $2$ | $8$ | $(1,7,6,5,8,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
