Explicit estimates on a theorem of Shioda concerning the ranksof curves given by y2=x3-a2x+m2

Nnumerous papers in the literature contain results on the ranks of ellipticcurves which generalize a theorem of Brown and Myers ony2=x3-x+m2in variousways. We have recently proven an effective version of a generalization which subsumesmost of what has appeared in the literature on this topic, although the bounds have notbeen worked out explicitly. We consider here the subfamilyy2=x3-a2x+m2, giveexplicit lower bounds formin terms ofafor such a curve to have rank at least 2, andshow how these lower bounds are influenced by theabcconjecture