Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22\cdot 29 + 25\cdot 29^{2} + 29^{3} + 24\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 23\cdot 29 + 10\cdot 29^{2} + 9\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 27\cdot 29 + 22\cdot 29^{2} + 27\cdot 29^{3} + 5\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 14\cdot 29 + 27\cdot 29^{2} + 27\cdot 29^{3} + 18\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)(2,3)$ |
| $(1,2,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
