Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 25\cdot 61 + 3\cdot 61^{2} + 7\cdot 61^{3} + 46\cdot 61^{4} + 54\cdot 61^{5} +O\left(61^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 61 + 59\cdot 61^{2} + 36\cdot 61^{3} + 9\cdot 61^{4} + 18\cdot 61^{5} + 19\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 26\cdot 61 + 26\cdot 61^{2} + 37\cdot 61^{3} + 15\cdot 61^{4} + 49\cdot 61^{5} + 46\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 50\cdot 61 + 61^{2} + 38\cdot 61^{3} + 12\cdot 61^{4} + 30\cdot 61^{5} + 38\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 33\cdot 61 + 19\cdot 61^{2} + 6\cdot 61^{3} + 52\cdot 61^{4} + 40\cdot 61^{5} + 47\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 9\cdot 61 + 14\cdot 61^{2} + 36\cdot 61^{3} + 15\cdot 61^{4} + 4\cdot 61^{5} + 5\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 + 36\cdot 61 + 41\cdot 61^{2} + 40\cdot 61^{3} + 47\cdot 61^{4} + 32\cdot 61^{5} + 16\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 54 + 60\cdot 61 + 16\cdot 61^{2} + 41\cdot 61^{3} + 44\cdot 61^{4} + 13\cdot 61^{5} + 8\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,4)(2,3,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
