Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12\cdot 29 + 25\cdot 29^{2} + 3\cdot 29^{4} + 3\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 3\cdot 29 + 25\cdot 29^{2} + 10\cdot 29^{3} + 11\cdot 29^{4} + 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 11\cdot 29 + 9\cdot 29^{2} + 16\cdot 29^{3} + 19\cdot 29^{4} + 26\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 + 12\cdot 29 + 12\cdot 29^{2} + 26\cdot 29^{3} + 7\cdot 29^{4} + 6\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 19\cdot 29 + 14\cdot 29^{2} + 3\cdot 29^{3} + 16\cdot 29^{4} + 20\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,5,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,2,5,3,4)$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,5,4,2,3)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,3,2,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,4,3,5,2)$ | $\zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
