Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 43\cdot 59 + 18\cdot 59^{2} + 8\cdot 59^{3} + 10\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 32\cdot 59 + 12\cdot 59^{2} + 32\cdot 59^{3} + 2\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 27\cdot 59 + 25\cdot 59^{2} + 51\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 39\cdot 59 + 45\cdot 59^{2} + 6\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 40\cdot 59 + 9\cdot 59^{2} + 46\cdot 59^{3} + 18\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 + 2\cdot 59 + 18\cdot 59^{2} + 31\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 + 15\cdot 59 + 2\cdot 59^{2} + 26\cdot 59^{3} + 18\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 51 + 35\cdot 59 + 44\cdot 59^{2} + 39\cdot 59^{3} + 6\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,6,2,5)(3,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
