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 + 6\cdot 19^{2} + 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 15\cdot 19 + 14\cdot 19^{2} + 18\cdot 19^{3} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 10\cdot 19^{2} + 19^{3} + 12\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 17\cdot 19 + 6\cdot 19^{2} + 17\cdot 19^{3} + 4\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,3)(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
