It is the fiber at t = 35033/156392 of Elkies' Z/4Z fibration E1 of the singular K3 surface of discriminant −163. The fiber also lies on the rank-5 subfamily, found by Dujella-Peral, defined by the condition 199664t² + 104728t − 122161 = □, parametrized by t = (−u² − 330u + 94936)/(u² + 199664), at u = 7052/65. The curve was found with the assistance of Claude, based on our old PARI/GP program.
dujella ·