A rank 3 family of elliptic curves of arbitrary \(j\)-invariant (Q1293753)

Let \(K\) be a number field. For each \(j_0 \in K\), the author considers the problem of finding an infinite number of elliptic curves with \(j\)-invariant equal to \(j_0\) and with ``high rank'' over \(K\). He constructs, for each \(j\)-invariant, infinitely many elliptic curves defined over \({\mathbb Q}(i)\) with rank at least \(3\), giving explicitly \(3\) linearly independent points, \(2\) of which are defined over \(\mathbb Q\).