Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57\cdot 61 + 43\cdot 61^{2} + 31\cdot 61^{3} + 51\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 10\cdot 61 + 6\cdot 61^{2} + 34\cdot 61^{3} + 48\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 30\cdot 61 + 11\cdot 61^{2} + 21\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 52\cdot 61 + 40\cdot 61^{2} + 38\cdot 61^{3} + 22\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 47\cdot 61 + 33\cdot 61^{2} + 35\cdot 61^{3} + 12\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 40\cdot 61 + 42\cdot 61^{2} + 53\cdot 61^{3} + 29\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 13\cdot 61 + 15\cdot 61^{2} + 51\cdot 61^{3} + 20\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 53 + 53\cdot 61 + 49\cdot 61^{2} + 38\cdot 61^{3} + 21\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,4)(3,7)(6,8)$ |
| $(1,6,5,8)(2,7,4,3)$ |
| $(1,2,5,4)(3,8,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ |
| $2$ | $4$ | $(1,2,5,4)(3,8,7,6)$ | $0$ |
| $2$ | $4$ | $(1,6,5,8)(2,7,4,3)$ | $0$ |
| $2$ | $4$ | $(1,7,5,3)(2,8,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
