Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 39\cdot 59 + 50\cdot 59^{2} + 27\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 32\cdot 59 + 29\cdot 59^{2} + 10\cdot 59^{3} + 47\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 48 + 42\cdot 59 + 59^{2} + 30\cdot 59^{3} + 24\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 54\cdot 59 + 53\cdot 59^{2} + 27\cdot 59^{3} + 47\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 + 34\cdot 59 + 15\cdot 59^{2} + 46\cdot 59^{3} + 21\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 35\cdot 59 + 39\cdot 59^{2} + 12\cdot 59^{3} + 29\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 56 + 7\cdot 59 + 12\cdot 59^{2} + 31\cdot 59^{3} + 24\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 46\cdot 59 + 32\cdot 59^{2} + 49\cdot 59^{3} + 57\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,5)(3,7)(6,8)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,6,7)(2,4,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
