y^2 = x(x+1)(x+103^2), from the Z/2Z x Z/4Z family y^2 = x(x+r^2)(x+s^2) with (r, s) = (1, 103); the point (rs, rs(r+s)) has order 4. Rank exactly 3 (PARI ellrank bounds agree). No curve with this torsion and rank 3 exists in Cremona's tables (complete to conductor 500000; all 1737 such curves scanned) or in LMFDB; within the family this is the smallest conductor for coprime r < s <= 450.
David Renshaw ·