Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 10\cdot 41 + 34\cdot 41^{2} + 4\cdot 41^{3} + 18\cdot 41^{4} + 33\cdot 41^{5} + 3\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 29\cdot 41 + 35\cdot 41^{2} + 36\cdot 41^{3} + 31\cdot 41^{4} + 37\cdot 41^{5} + 39\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 17\cdot 41 + 31\cdot 41^{2} + 36\cdot 41^{3} + 23\cdot 41^{4} + 28\cdot 41^{5} + 10\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 32\cdot 41 + 30\cdot 41^{2} + 20\cdot 41^{3} + 31\cdot 41^{5} + 38\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 35\cdot 41 + 8\cdot 41^{2} + 18\cdot 41^{3} + 25\cdot 41^{5} + 4\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 3\cdot 41 + 20\cdot 41^{2} + 23\cdot 41^{3} + 8\cdot 41^{4} + 5\cdot 41^{5} + 39\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 38 + 9\cdot 41 + 22\cdot 41^{2} + 19\cdot 41^{3} + 31\cdot 41^{4} + 20\cdot 41^{5} + 40\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 39 + 24\cdot 41 + 21\cdot 41^{2} + 3\cdot 41^{3} + 8\cdot 41^{4} + 23\cdot 41^{5} + 27\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,6)(5,7)$ |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,4,2,7)(3,5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,2,7)(3,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
