Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 21\cdot 41 + 5\cdot 41^{2} + 41^{3} + 21\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 20\cdot 41 + 37\cdot 41^{2} + 34\cdot 41^{3} + 26\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 28\cdot 41 + 8\cdot 41^{2} + 13\cdot 41^{3} + 18\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 6\cdot 41 + 25\cdot 41^{2} + 13\cdot 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 12\cdot 41 + 30\cdot 41^{2} + 32\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 29\cdot 41 + 31\cdot 41^{2} + 17\cdot 41^{3} + 3\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 32 + 27\cdot 41 + 7\cdot 41^{2} + 12\cdot 41^{3} + 25\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 34 + 17\cdot 41 + 17\cdot 41^{2} + 38\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,3)(6,7)$ |
| $(1,5)(2,3)(4,8)(6,7)$ |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,8,5,4)(2,6,3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(2,3)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,8,5,4)(2,6,3,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,5,8)(2,7,3,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,5,7)(2,8,3,4)$ | $0$ |
| $2$ | $4$ | $(1,3,5,2)(4,6,8,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,4)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
