  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. The situation for A not
principally polarised remains unclear. For each k in {1,2,3,5,6,7,10,13} we
construct a non-simple non-principally polarised abelian surface over the
rationals whose Tate-Shafarevich group has order k times a square. To obtain
this result, we explore the invariance under isogeny of the Birch and
Swinnerton-Dyer conjecture.
