Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 21\cdot 29 + 5\cdot 29^{2} + 29^{3} + 20\cdot 29^{4} + 25\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 2\cdot 29 + 5\cdot 29^{2} + 19\cdot 29^{3} + 4\cdot 29^{4} + 20\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 17\cdot 29 + 23\cdot 29^{2} + 10\cdot 29^{3} + 7\cdot 29^{4} + 28\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 17\cdot 29 + 23\cdot 29^{2} + 26\cdot 29^{3} + 25\cdot 29^{4} + 12\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
