  Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a
rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an
algorithm to decide whether the Tate-Shafarevich group of the abelian surface B
has square order or order five times a square, assuming that we can find a
basis for the Mordell-Weil groups of both E_i, and that the Tate-Shafarevich
groups of the E_i are finite.
  We considered all pairs (E_1,E_2), with prescribed bounds on the conductor
and the coefficients on a minimal Weierstrass equation. In total we considered
around 20.0 million of abelian surfaces of which 49.16% have a Tate-Shafarevich
group of non-square order.
