Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 41 + 11\cdot 41^{2} + 28\cdot 41^{3} + 18\cdot 41^{4} + 6\cdot 41^{5} + 30\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 12\cdot 41 + 30\cdot 41^{2} + 40\cdot 41^{3} + 9\cdot 41^{4} + 8\cdot 41^{5} + 12\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 4\cdot 41 + 9\cdot 41^{2} + 8\cdot 41^{3} + 34\cdot 41^{4} + 10\cdot 41^{5} + 8\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 25\cdot 41 + 12\cdot 41^{2} + 20\cdot 41^{3} + 15\cdot 41^{4} + 28\cdot 41^{5} + 9\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 15\cdot 41 + 28\cdot 41^{2} + 20\cdot 41^{3} + 25\cdot 41^{4} + 12\cdot 41^{5} + 31\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 + 36\cdot 41 + 31\cdot 41^{2} + 32\cdot 41^{3} + 6\cdot 41^{4} + 30\cdot 41^{5} + 32\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 + 28\cdot 41 + 10\cdot 41^{2} + 31\cdot 41^{4} + 32\cdot 41^{5} + 28\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 37 + 39\cdot 41 + 29\cdot 41^{2} + 12\cdot 41^{3} + 22\cdot 41^{4} + 34\cdot 41^{5} + 10\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,5)(2,3,4,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
