Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 10\cdot 29 + 3\cdot 29^{2} + 8\cdot 29^{3} + 21\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 8\cdot 29 + 24\cdot 29^{2} + 5\cdot 29^{3} + 17\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 9\cdot 29 + 29^{2} + 15\cdot 29^{3} + 19\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
