Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 33\cdot 41 + 25\cdot 41^{2} + 3\cdot 41^{3} + 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 24\cdot 41 + 37\cdot 41^{2} + 6\cdot 41^{3} + 30\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 41 + 19\cdot 41^{2} + 26\cdot 41^{3} + 25\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 7\cdot 41 + 10\cdot 41^{2} + 11\cdot 41^{3} + 27\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 33\cdot 41 + 30\cdot 41^{2} + 29\cdot 41^{3} + 13\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 39\cdot 41 + 21\cdot 41^{2} + 14\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 + 16\cdot 41 + 3\cdot 41^{2} + 34\cdot 41^{3} + 10\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 35 + 7\cdot 41 + 15\cdot 41^{2} + 37\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2,6,4)(3,5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,6,4)(3,5,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
