Explicit models of genus 2 curves with split CM
  We outline a general algorithm for computing an explicit model over a number field of any curve of genus 2 whose (unpolarized) Jacobian is isomorphic to the product of two elliptic curves with CM by the same order in an imaginary quadratic field. We give the details and some examples for the case where the order has prime discriminant and class number one 
Fernando Rodriguez Villegas