Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 24\cdot 41 + 41^{2} + 10\cdot 41^{3} + 34\cdot 41^{4} + 8\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 29\cdot 41 + 21\cdot 41^{2} + 33\cdot 41^{3} + 41^{4} + 22\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 35\cdot 41^{2} + 4\cdot 41^{3} + 32\cdot 41^{4} + 10\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 2\cdot 41 + 17\cdot 41^{2} + 23\cdot 41^{3} + 3\cdot 41^{4} + 5\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 10\cdot 41 + 14\cdot 41^{2} + 13\cdot 41^{3} + 40\cdot 41^{4} + 7\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 28\cdot 41 + 23\cdot 41^{2} + 33\cdot 41^{3} + 13\cdot 41^{4} + 40\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 + 34\cdot 41 + 37\cdot 41^{3} + 7\cdot 41^{4} + 33\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 31 + 34\cdot 41 + 8\cdot 41^{2} + 8\cdot 41^{3} + 30\cdot 41^{4} + 35\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(1,2,6,3)(4,8,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,2,6,3)(4,8,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
