Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 50\cdot 61 + 10\cdot 61^{2} + 25\cdot 61^{3} + 28\cdot 61^{4} + 40\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 29\cdot 61 + 21\cdot 61^{2} + 27\cdot 61^{3} + 38\cdot 61^{4} + 7\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 47\cdot 61 + 32\cdot 61^{2} + 57\cdot 61^{3} + 14\cdot 61^{4} + 25\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 21\cdot 61 + 22\cdot 61^{2} + 28\cdot 61^{3} + 53\cdot 61^{4} + 27\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 39\cdot 61 + 38\cdot 61^{2} + 32\cdot 61^{3} + 7\cdot 61^{4} + 33\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 13\cdot 61 + 28\cdot 61^{2} + 3\cdot 61^{3} + 46\cdot 61^{4} + 35\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 31\cdot 61 + 39\cdot 61^{2} + 33\cdot 61^{3} + 22\cdot 61^{4} + 53\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 10\cdot 61 + 50\cdot 61^{2} + 35\cdot 61^{3} + 32\cdot 61^{4} + 20\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
