Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 54\cdot 61^{2} + 5\cdot 61^{3} + 53\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 19\cdot 61 + 41\cdot 61^{2} + 56\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 11\cdot 61 + 46\cdot 61^{2} + 46\cdot 61^{3} + 59\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 19\cdot 61 + 35\cdot 61^{2} + 43\cdot 61^{3} + 60\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 41\cdot 61 + 25\cdot 61^{2} + 17\cdot 61^{3} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 49\cdot 61 + 14\cdot 61^{2} + 14\cdot 61^{3} + 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 41\cdot 61 + 19\cdot 61^{2} + 4\cdot 61^{3} + 4\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 43 + 60\cdot 61 + 6\cdot 61^{2} + 55\cdot 61^{3} + 7\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,7)(2,8,6,4)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
