Found by a conductor-oriented search in the standard Z/2Z x Z/4Z family \(y^2=x(x+u^2)(x+v^2)\), using conditions \(u+v=d w^2\) with small squarefree \(d\). Here \((u,v)=(8791,11900)\) and \(u+v=20691=19\cdot33^2\). PARI/GP 2.17.4 gives exact rank 5 and torsion Z/2Z x Z/4Z. Conductor \(N=10196321216235\). This improves my preceding rank-5 submission of conductor \(38019384450552\).
jesper-petersen ·