Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 9\cdot 41 + 10\cdot 41^{2} + 11\cdot 41^{3} + 17\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 6\cdot 41 + 8\cdot 41^{2} + 3\cdot 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 20\cdot 41 + 36\cdot 41^{2} + 18\cdot 41^{3} + 33\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 5\cdot 41 + 27\cdot 41^{2} + 7\cdot 41^{3} + 17\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 5\cdot 41 + 6\cdot 41^{2} + 29\cdot 41^{3} + 12\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 + 9\cdot 41 + 12\cdot 41^{2} + 26\cdot 41^{3} + 18\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 16\cdot 41 + 32\cdot 41^{2} + 36\cdot 41^{3} + 28\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 39 + 8\cdot 41 + 31\cdot 41^{2} + 30\cdot 41^{3} + 21\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,6,7)(2,3,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
