Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 13\cdot 19 + 13\cdot 19^{2} + 9\cdot 19^{3} + 10\cdot 19^{4} + 8\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 17\cdot 19 + 10\cdot 19^{2} + 9\cdot 19^{3} + 7\cdot 19^{4} + 2\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 19 + 8\cdot 19^{2} + 9\cdot 19^{3} + 11\cdot 19^{4} + 16\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 5\cdot 19 + 5\cdot 19^{2} + 9\cdot 19^{3} + 8\cdot 19^{4} + 10\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,3,4,2)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
