On the equation y2=x(x−2m)(x+q−2m)

AbstractConsider a family of elliptic curves Eq,m:y2=x(x−2m)(x+q−2m), where q is an odd prime satisfying q−2m>0. In a case q−2m is a prime, we give fairly complete formula for the rank, and describe an elementary method to search for non-trivial points. In general case we can prove that either the rank or 2-part of the Tate–Shafarevich group can be arbitrarily large. We also prove (under reasonable assumptions) that for any partition k=l+n into non-negative integers there are pairwise nonisogeneous elliptic curves E1,…,Ek among Eq,m's such that for a positive proportion of prime quadratic twists by p we have: rankE1(p)=⋯=rankEl(p)=0 and rankEl+1(p)=⋯=rankEk(p)=1. We prove explicit estimates for the canonical height on (quadratic twists of) Eq,m (in a case q−2m is a prime) and include a list of values of the analytic order of