Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 57\cdot 61 + 24\cdot 61^{2} + 10\cdot 61^{3} + 55\cdot 61^{4} + 24\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 59\cdot 61 + 12\cdot 61^{2} + 39\cdot 61^{3} + 20\cdot 61^{4} + 46\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 42\cdot 61 + 33\cdot 61^{2} + 61^{3} + 41\cdot 61^{4} + 38\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 7\cdot 61 + 41\cdot 61^{2} + 31\cdot 61^{3} + 9\cdot 61^{4} + 11\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 50\cdot 61 + 5\cdot 61^{2} + 40\cdot 61^{3} + 47\cdot 61^{4} + 23\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 21\cdot 61 + 41\cdot 61^{2} + 48\cdot 61^{3} + 23\cdot 61^{4} + 48\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 48 + 53\cdot 61 + 49\cdot 61^{2} + 22\cdot 61^{3} + 56\cdot 61^{4} + 24\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 53 + 12\cdot 61 + 34\cdot 61^{2} + 49\cdot 61^{3} + 50\cdot 61^{4} + 25\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,7,8)(3,6,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,2,7,8)(3,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
