Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 7\cdot 41 + 32\cdot 41^{2} + 23\cdot 41^{3} + 19\cdot 41^{4} + 16\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 20\cdot 41 + 12\cdot 41^{2} + 22\cdot 41^{4} + 25\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 11\cdot 41 + 25\cdot 41^{2} + 6\cdot 41^{3} + 30\cdot 41^{4} + 8\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 35\cdot 41 + 2\cdot 41^{2} + 16\cdot 41^{3} + 13\cdot 41^{4} + 13\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 29\cdot 41 + 23\cdot 41^{2} + 22\cdot 41^{3} + 24\cdot 41^{4} + 34\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 35\cdot 41 + 4\cdot 41^{2} + 13\cdot 41^{3} + 23\cdot 41^{4} + 28\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 11\cdot 41 + 4\cdot 41^{2} + 6\cdot 41^{3} + 26\cdot 41^{4} + 32\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 39 + 11\cdot 41 + 17\cdot 41^{2} + 34\cdot 41^{3} + 4\cdot 41^{4} + 4\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,6)(2,4,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
