Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 39\cdot 59 + 51\cdot 59^{2} + 54\cdot 59^{3} + 18\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 19\cdot 59 + 58\cdot 59^{2} + 50\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 14\cdot 59 + 8\cdot 59^{2} + 42\cdot 59^{3} + 54\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 58\cdot 59 + 18\cdot 59^{2} + 6\cdot 59^{3} + 13\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 32\cdot 59 + 2\cdot 59^{2} + 26\cdot 59^{3} + 24\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 35\cdot 59 + 17\cdot 59^{2} + 17\cdot 59^{3} + 49\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 + 25\cdot 59 + 5\cdot 59^{2} + 2\cdot 59^{3} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 + 10\cdot 59 + 14\cdot 59^{2} + 36\cdot 59^{3} + 39\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,4,8)(2,3,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
