Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 55\cdot 61 + 58\cdot 61^{2} + 28\cdot 61^{3} + 19\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 56\cdot 61 + 11\cdot 61^{2} + 58\cdot 61^{3} + 4\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 30\cdot 61 + 60\cdot 61^{2} + 22\cdot 61^{3} + 59\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 23\cdot 61 + 44\cdot 61^{2} + 52\cdot 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 + 6\cdot 61 + 60\cdot 61^{2} + 10\cdot 61^{3} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 45\cdot 61 + 12\cdot 61^{2} + 54\cdot 61^{3} + 54\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 33\cdot 61 + 51\cdot 61^{2} + 19\cdot 61^{3} + 28\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 60 + 53\cdot 61 + 4\cdot 61^{2} + 57\cdot 61^{3} + 25\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,5,8)(3,7,4,6)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,2,5,8)(3,7,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
