Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 36\cdot 59 + 57\cdot 59^{2} + 6\cdot 59^{3} + 41\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 33\cdot 59 + 56\cdot 59^{2} + 7\cdot 59^{3} + 21\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 35\cdot 59 + 50\cdot 59^{2} + 21\cdot 59^{3} + 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 56\cdot 59 + 53\cdot 59^{2} + 4\cdot 59^{3} + 49\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 2\cdot 59 + 5\cdot 59^{2} + 54\cdot 59^{3} + 9\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 23\cdot 59 + 8\cdot 59^{2} + 37\cdot 59^{3} + 57\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 25\cdot 59 + 2\cdot 59^{2} + 51\cdot 59^{3} + 37\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 22\cdot 59 + 59^{2} + 52\cdot 59^{3} + 17\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,2,6)(3,8,5,7)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,2,6)(3,8,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
