Examples of non-simple abelian surfaces over the rationals with non-square order Tate–Shafarevich group

Let A be an abelian surface over a fixed number field. If A is principally polarised, then it is known that the order of the Tate–Shafarevich group of A   must, if finite, be a square or twice a square. For each k∈{1,2,3,5,6,7,10,13}k∈{1,2,3,5,6,7,10,13} we construct a non-simple non-principally polarised abelian surface B/QB/Q having Tate–Shafarevich group of order k times a square. To obtain this result, we explore the invariance under isogeny of the Birch and Swinnerton–Dyer conjecture.