Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 10\cdot 29 + 13\cdot 29^{2} + 6\cdot 29^{3} + 18\cdot 29^{4} + 26\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 13\cdot 29 + 20\cdot 29^{2} + 3\cdot 29^{3} + 24\cdot 29^{4} + 12\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 27\cdot 29 + 26\cdot 29^{2} + 7\cdot 29^{3} + 24\cdot 29^{4} + 6\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 28\cdot 29 + 2\cdot 29^{2} + 20\cdot 29^{3} + 18\cdot 29^{4} + 12\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 26\cdot 29^{2} + 8\cdot 29^{3} + 10\cdot 29^{4} + 16\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 29 + 2\cdot 29^{2} + 21\cdot 29^{3} + 4\cdot 29^{4} + 22\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 + 15\cdot 29 + 8\cdot 29^{2} + 25\cdot 29^{3} + 4\cdot 29^{4} + 16\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 28 + 18\cdot 29 + 15\cdot 29^{2} + 22\cdot 29^{3} + 10\cdot 29^{4} + 2\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
