Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 44\cdot 61 + 33\cdot 61^{2} + 32\cdot 61^{3} + 17\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 6\cdot 61 + 52\cdot 61^{2} + 3\cdot 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 59\cdot 61 + 15\cdot 61^{2} + 45\cdot 61^{3} + 58\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 21\cdot 61 + 55\cdot 61^{2} + 4\cdot 61^{3} + 58\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 14\cdot 61 + 26\cdot 61^{2} + 29\cdot 61^{3} + 51\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 35\cdot 61 + 59\cdot 61^{2} + 6\cdot 61^{3} + 16\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 50\cdot 61 + 41\cdot 61^{2} + 19\cdot 61^{3} + 57\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 53 + 12\cdot 61 + 20\cdot 61^{2} + 40\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,7)(3,6,5,8)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,4,7)(3,6,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
