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