Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 12\cdot 41 + 3\cdot 41^{2} + 5\cdot 41^{3} + 24\cdot 41^{4} + 9\cdot 41^{5} + 18\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 35\cdot 41 + 25\cdot 41^{2} + 25\cdot 41^{3} + 15\cdot 41^{4} + 16\cdot 41^{5} + 34\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 2\cdot 41 + 17\cdot 41^{2} + 35\cdot 41^{3} + 9\cdot 41^{4} + 20\cdot 41^{5} + 28\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 15\cdot 41 + 41^{2} + 26\cdot 41^{3} + 39\cdot 41^{4} + 13\cdot 41^{5} + 37\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 25\cdot 41 + 39\cdot 41^{2} + 14\cdot 41^{3} + 41^{4} + 27\cdot 41^{5} + 3\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 + 38\cdot 41 + 23\cdot 41^{2} + 5\cdot 41^{3} + 31\cdot 41^{4} + 20\cdot 41^{5} + 12\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 + 5\cdot 41 + 15\cdot 41^{2} + 15\cdot 41^{3} + 25\cdot 41^{4} + 24\cdot 41^{5} + 6\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 36 + 28\cdot 41 + 37\cdot 41^{2} + 35\cdot 41^{3} + 16\cdot 41^{4} + 31\cdot 41^{5} + 22\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,6)(2,4,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
