  The purpose of the present article is to show how the modular method together
with different techniques can be used to prove non-existence of primitive
non-trivial solutions of the equation $x^2+dy^6=z^p$ for square-free values $1
\le d \le 20$ following the approach of [PT]. The main innovation is to make
use of the symplectic argument over ramified extensions to discard solutions,
together with a multi-Frey approach to deduce large image of Galois
representations.
