Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 28\cdot 29 + 7\cdot 29^{2} + 3\cdot 29^{3} + 20\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 9\cdot 29 + 26\cdot 29^{2} + 19\cdot 29^{3} + 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 21\cdot 29 + 14\cdot 29^{2} + 10\cdot 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 2\cdot 29 + 4\cdot 29^{2} + 27\cdot 29^{3} + 26\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 26\cdot 29 + 24\cdot 29^{2} + 29^{3} + 2\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 7\cdot 29 + 14\cdot 29^{2} + 18\cdot 29^{3} + 12\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 + 19\cdot 29 + 2\cdot 29^{2} + 9\cdot 29^{3} + 27\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 + 21\cdot 29^{2} + 25\cdot 29^{3} + 8\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,3)(5,6,8,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
