A family of elliptic curves of large rank

Fix a finite field k, a positive integer d relatively prime to the characteristic of k, and an element a of k. In this article we study the elliptic curve E with equation x(x−1)(y−a)=tdy(y−1)(x−a) over k(t). We find a formula for the L-function of E for d=q+1, and we show that for all d the rank of E over is bounded below by d. This is notable because previous studies of similar curves have found large ranks only for much more restricted d. Further, we provide an explicit lower bound on the rank of E over k(t) which shows that the rank of E is unbounded as d varies even over k(t).