Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 8\cdot 41 + 32\cdot 41^{2} + 40\cdot 41^{3} + 13\cdot 41^{4} + 31\cdot 41^{5} + 2\cdot 41^{6} + 28\cdot 41^{7} + 28\cdot 41^{8} + 38\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 14\cdot 41 + 34\cdot 41^{2} + 4\cdot 41^{3} + 24\cdot 41^{4} + 10\cdot 41^{5} + 7\cdot 41^{6} + 8\cdot 41^{7} + 25\cdot 41^{8} + 16\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 30\cdot 41 + 13\cdot 41^{2} + 2\cdot 41^{3} + 29\cdot 41^{4} + 40\cdot 41^{5} + 14\cdot 41^{6} + 41^{7} + 34\cdot 41^{8} + 32\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 29\cdot 41 + 15\cdot 41^{2} + 37\cdot 41^{3} + 17\cdot 41^{4} + 32\cdot 41^{5} + 18\cdot 41^{6} + 2\cdot 41^{7} + 14\cdot 41^{8} +O\left(41^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 + 11\cdot 41 + 25\cdot 41^{2} + 3\cdot 41^{3} + 23\cdot 41^{4} + 8\cdot 41^{5} + 22\cdot 41^{6} + 38\cdot 41^{7} + 26\cdot 41^{8} + 40\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 10\cdot 41 + 27\cdot 41^{2} + 38\cdot 41^{3} + 11\cdot 41^{4} + 26\cdot 41^{6} + 39\cdot 41^{7} + 6\cdot 41^{8} + 8\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 36 + 26\cdot 41 + 6\cdot 41^{2} + 36\cdot 41^{3} + 16\cdot 41^{4} + 30\cdot 41^{5} + 33\cdot 41^{6} + 32\cdot 41^{7} + 15\cdot 41^{8} + 24\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 37 + 32\cdot 41 + 8\cdot 41^{2} + 27\cdot 41^{4} + 9\cdot 41^{5} + 38\cdot 41^{6} + 12\cdot 41^{7} + 12\cdot 41^{8} + 2\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(2,7)(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,5,7,6,8,4,2,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
