Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 38\cdot 61 + 23\cdot 61^{2} + 22\cdot 61^{3} + 55\cdot 61^{4} + 23\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 30\cdot 61 + 42\cdot 61^{2} + 40\cdot 61^{3} + 55\cdot 61^{4} + 30\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 49\cdot 61 + 19\cdot 61^{2} + 56\cdot 61^{3} + 4\cdot 61^{4} + 39\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 4\cdot 61 + 36\cdot 61^{2} + 2\cdot 61^{3} + 6\cdot 61^{4} + 28\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 34\cdot 61 + 7\cdot 61^{2} + 45\cdot 61^{3} + 43\cdot 61^{4} + 3\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 + 52\cdot 61 + 35\cdot 61^{2} + 33\cdot 61^{3} + 16\cdot 61^{4} + 59\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 + 35\cdot 61 + 13\cdot 61^{2} + 28\cdot 61^{3} + 30\cdot 61^{4} + 40\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 + 59\cdot 61 + 3\cdot 61^{2} + 15\cdot 61^{3} + 31\cdot 61^{4} + 18\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,4)(5,8,6,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
