Ranks and integer points on elliptic curves induced by Fibonacci triples

Let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, respectively, and consider \[ E_k:\qquad y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1). \] These elliptic curves arise naturally from the regular Diophantine triples \[ \{F_{2k},F_{2k+2},F_{2k+4}\}. \] For odd $k$, we exhibit the rational p