Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 39\cdot 41 + 28\cdot 41^{2} + 2\cdot 41^{3} + 7\cdot 41^{4} + 19\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 16\cdot 41 + 29\cdot 41^{2} + 32\cdot 41^{3} + 39\cdot 41^{4} + 2\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 14\cdot 41 + 30\cdot 41^{2} + 4\cdot 41^{3} + 36\cdot 41^{4} + 10\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 10\cdot 41 + 22\cdot 41^{2} + 6\cdot 41^{3} + 5\cdot 41^{4} + 18\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 30\cdot 41 + 18\cdot 41^{2} + 34\cdot 41^{3} + 35\cdot 41^{4} + 22\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 + 26\cdot 41 + 10\cdot 41^{2} + 36\cdot 41^{3} + 4\cdot 41^{4} + 30\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 38 + 24\cdot 41 + 11\cdot 41^{2} + 8\cdot 41^{3} + 41^{4} + 38\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 + 41 + 12\cdot 41^{2} + 38\cdot 41^{3} + 33\cdot 41^{4} + 21\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
