Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 11\cdot 61 + 39\cdot 61^{2} + 57\cdot 61^{3} + 51\cdot 61^{4} + 30\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 34\cdot 61^{2} + 57\cdot 61^{3} + 27\cdot 61^{4} + 56\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 19\cdot 61 + 53\cdot 61^{2} + 33\cdot 61^{3} + 25\cdot 61^{4} + 18\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 52\cdot 61 + 12\cdot 61^{2} + 27\cdot 61^{3} + 59\cdot 61^{4} + 16\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 + 8\cdot 61 + 48\cdot 61^{2} + 33\cdot 61^{3} + 61^{4} + 44\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 41\cdot 61 + 7\cdot 61^{2} + 27\cdot 61^{3} + 35\cdot 61^{4} + 42\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 60\cdot 61 + 26\cdot 61^{2} + 3\cdot 61^{3} + 33\cdot 61^{4} + 4\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 + 49\cdot 61 + 21\cdot 61^{2} + 3\cdot 61^{3} + 9\cdot 61^{4} + 30\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,5)(2,3,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
