Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 25\cdot 59 + 4\cdot 59^{2} + 34\cdot 59^{3} + 13\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 5\cdot 59 + 10\cdot 59^{2} + 55\cdot 59^{3} + 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 32\cdot 59 + 22\cdot 59^{2} + 43\cdot 59^{3} + 16\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 44\cdot 59 + 14\cdot 59^{2} + 17\cdot 59^{3} + 41\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 14\cdot 59 + 44\cdot 59^{2} + 41\cdot 59^{3} + 17\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 43 + 26\cdot 59 + 36\cdot 59^{2} + 15\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 + 53\cdot 59 + 48\cdot 59^{2} + 3\cdot 59^{3} + 57\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 33\cdot 59 + 54\cdot 59^{2} + 24\cdot 59^{3} + 45\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,3,7)(2,8,4,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,5,3,7)(2,8,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
