Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 4\cdot 19 + 7\cdot 19^{2} + 12\cdot 19^{3} + 9\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 19 + 18\cdot 19^{2} + 17\cdot 19^{3} + 6\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 12\cdot 19 + 12\cdot 19^{2} + 7\cdot 19^{3} + 2\cdot 19^{4} +O\left(19^{ 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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,2,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
