Great question. There's a subtle point the article glosses over "... with coefficients in K." If you have two elliptic curves that just happen to be cyclic gr

Great question. There's a subtle point the article glosses over "... with coefficients in K." If you have two elliptic curves that just happen to be cyclic groups over finite fields, you can construct homomorphisms between them with finite kernels. But, you won't have any hope of finding morphisms unless those finite fields are the same. (same characteristic? It's been a while) In less fancy language: isogenies are homomorphisms you can describe with simple formulas involving (x, y) coordinates.