Quartic Diophantine equation in three variables

Question

Is there a positive integer solution to the quartic Diophantine equation? $$x^4-4x^2y^2+8y^4=z^2$$

Cf. Yiu,"Recreational Mathematics" Chap. 6.2 pp. 50/360

Sinha, T. N. Two simultaneous diophantine equations. Math. Student 33 (1965), 59-61

Answer 1

The answer is NO.

The quartic is birationally equivalent to the elliptic curve \begin{equation*} v^2=u(u^2+2u-1) \end{equation*} with \begin{equation*} \frac{x}{y}=\frac{v}{u} \end{equation*}

Pari-GP gives the torsion subgroup of the curve as having only one finite point $(0,0)$, whilst Denis Simon's $\mathbf{ellrank}$ code gives the rank as $0$.

Thus, the only rational point on the curve is $(0,0)$ which does not, clearly, give a solution.