y^2=x^3+Dz^12,

I have my student working on finding primitive integer solutions to some Diophantine equations of the form $C:y^2=x^3+Dz^{12}$. This is a genus two curve (in the appropriate weighted projective space), and it is a double cover of an elliptic curve, so computation of the Mordell-Weil rank is usually pretty straightforward. Say we find a $D$ so that the elliptic curve $y^2=x^3+D$ has rank $2$, but we suspect that there are no non-primitive solutions there. Can we prove this? I suspect that a simple a Mordell-Weil sieving should give us the desired result, and it seems pretty successful so far. However, it seems the trivial solutions
$(1,1,0)$ is causing some trouble for us. Right now, we're trying to figure out if there is a way of getting pass this problem.