Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 15\cdot 43 + 9\cdot 43^{2} + 23\cdot 43^{3} + 8\cdot 43^{4} + 29\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 3\cdot 43 + 3\cdot 43^{2} + 40\cdot 43^{3} + 7\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 15\cdot 43 + 13\cdot 43^{2} + 27\cdot 43^{4} + 26\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 + 42\cdot 43 + 31\cdot 43^{2} + 2\cdot 43^{3} + 5\cdot 43^{4} + 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 + 32\cdot 43 + 35\cdot 43^{2} + 20\cdot 43^{3} + 35\cdot 43^{4} + 32\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 30\cdot 43 + 35\cdot 43^{2} + 11\cdot 43^{3} + 31\cdot 43^{4} + 26\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 32 + 41\cdot 43 + 5\cdot 43^{2} + 33\cdot 43^{3} + 32\cdot 43^{4} + 32\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 + 34\cdot 43 + 36\cdot 43^{2} + 39\cdot 43^{3} + 30\cdot 43^{4} + 15\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(3,5)(4,8)$ |
| $(1,7,3,5)(2,8,6,4)$ |
| $(1,3)(2,6)(4,8)(5,7)$ |
| $(1,4,3,8)(2,7,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $-2$ |
| $4$ | $2$ | $(1,7)(3,5)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,7,3,5)(2,8,6,4)$ | $0$ |
| $4$ | $4$ | $(1,4,3,8)(2,7,6,5)$ | $0$ |
| $2$ | $8$ | $(1,8,7,6,3,4,5,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,2,3,8,5,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
