Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 56\cdot 59 + 31\cdot 59^{2} + 6\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 35\cdot 59 + 54\cdot 59^{2} + 48\cdot 59^{3} + 58\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 54\cdot 59 + 47\cdot 59^{2} + 46\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 35\cdot 59 + 43\cdot 59^{2} + 34\cdot 59^{3} + 38\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 24\cdot 59 + 44\cdot 59^{2} + 30\cdot 59^{3} + 19\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 + 14\cdot 59 + 18\cdot 59^{2} + 46\cdot 59^{3} + 37\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 19\cdot 59 + 41\cdot 59^{2} + 49\cdot 59^{3} + 17\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 55\cdot 59 + 12\cdot 59^{2} + 18\cdot 59^{3} + 49\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,8)(3,7)(4,6)$ |
| $(1,3)(2,4)(5,6)(7,8)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,8)(3,7)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,2,6)(3,5,4,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
