Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 20\cdot 29 + 21\cdot 29^{2} + 24\cdot 29^{3} + 13\cdot 29^{4} + 24\cdot 29^{5} + 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 6\cdot 29 + 9\cdot 29^{2} + 6\cdot 29^{3} + 20\cdot 29^{4} + 25\cdot 29^{5} + 14\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 22\cdot 29 + 19\cdot 29^{2} + 22\cdot 29^{3} + 8\cdot 29^{4} + 3\cdot 29^{5} + 14\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 8\cdot 29 + 7\cdot 29^{2} + 4\cdot 29^{3} + 15\cdot 29^{4} + 4\cdot 29^{5} + 27\cdot 29^{6} +O\left(29^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
