Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 14\cdot 59 + 11\cdot 59^{2} + 53\cdot 59^{3} + 52\cdot 59^{4} + 17\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 56\cdot 59 + 54\cdot 59^{2} + 32\cdot 59^{3} + 50\cdot 59^{4} + 55\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 46\cdot 59 + 14\cdot 59^{2} + 52\cdot 59^{3} + 29\cdot 59^{4} + 2\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 57\cdot 59 + 21\cdot 59^{2} + 20\cdot 59^{3} + 15\cdot 59^{4} + 17\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 59 + 37\cdot 59^{2} + 38\cdot 59^{3} + 43\cdot 59^{4} + 41\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 12\cdot 59 + 44\cdot 59^{2} + 6\cdot 59^{3} + 29\cdot 59^{4} + 56\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 54 + 2\cdot 59 + 4\cdot 59^{2} + 26\cdot 59^{3} + 8\cdot 59^{4} + 3\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 + 44\cdot 59 + 47\cdot 59^{2} + 5\cdot 59^{3} + 6\cdot 59^{4} + 41\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,2,5)(4,8,6,7)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,5)(4,8,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
