Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 58\cdot 61 + 7\cdot 61^{2} + 24\cdot 61^{3} + 19\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 50\cdot 61 + 26\cdot 61^{2} + 23\cdot 61^{3} + 60\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 61 + 31\cdot 61^{2} + 61^{3} + 15\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 14\cdot 61 + 54\cdot 61^{2} + 59\cdot 61^{3} + 23\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 46\cdot 61 + 6\cdot 61^{2} + 61^{3} + 37\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 59\cdot 61 + 29\cdot 61^{2} + 59\cdot 61^{3} + 45\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 10\cdot 61 + 34\cdot 61^{2} + 37\cdot 61^{3} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 2\cdot 61 + 53\cdot 61^{2} + 36\cdot 61^{3} + 41\cdot 61^{4} +O\left(61^{ 5 }\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.
