Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 33\cdot 41^{2} + 38\cdot 41^{3} + 26\cdot 41^{4} + 6\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 38\cdot 41 + 5\cdot 41^{2} + 9\cdot 41^{3} + 8\cdot 41^{4} + 10\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 40\cdot 41 + 36\cdot 41^{2} + 10\cdot 41^{3} + 17\cdot 41^{4} + 26\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 37\cdot 41 + 9\cdot 41^{2} + 22\cdot 41^{3} + 39\cdot 41^{4} + 29\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 3\cdot 41 + 31\cdot 41^{2} + 18\cdot 41^{3} + 41^{4} + 11\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 4\cdot 41^{2} + 30\cdot 41^{3} + 23\cdot 41^{4} + 14\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 + 2\cdot 41 + 35\cdot 41^{2} + 31\cdot 41^{3} + 32\cdot 41^{4} + 30\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 + 40\cdot 41 + 7\cdot 41^{2} + 2\cdot 41^{3} + 14\cdot 41^{4} + 34\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,7,4,6)(2,5,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
