On the equation y2=x(x−m2)(x+q−m2)

Consider a family of elliptic curves , where q is an odd prime satisfying q−m2>0. In a case q−m2 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: and . We prove explicit estimates for the canonical height on (quadratic twists of) Eq,m (in a case q−m2 is a prime) and include a list of values of the analytic order of .