Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 18\cdot 61 + 5\cdot 61^{2} + 2\cdot 61^{3} + 40\cdot 61^{4} + 13\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 21\cdot 61 + 33\cdot 61^{2} + 9\cdot 61^{3} + 31\cdot 61^{4} + 15\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 50\cdot 61 + 20\cdot 61^{2} + 52\cdot 61^{3} + 34\cdot 61^{4} + 11\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 40\cdot 61 + 28\cdot 61^{2} + 11\cdot 61^{3} + 27\cdot 61^{4} + 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 + 20\cdot 61 + 32\cdot 61^{2} + 49\cdot 61^{3} + 33\cdot 61^{4} + 59\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 10\cdot 61 + 40\cdot 61^{2} + 8\cdot 61^{3} + 26\cdot 61^{4} + 49\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 46 + 39\cdot 61 + 27\cdot 61^{2} + 51\cdot 61^{3} + 29\cdot 61^{4} + 45\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 42\cdot 61 + 55\cdot 61^{2} + 58\cdot 61^{3} + 20\cdot 61^{4} + 47\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
