An Explicit Manin-Dem'janenko theorem in elliptic curves

Let C be a curve of genus at least 2 imbedded in a product of elliptic curves. We give an explicit upper bound for the points in the intersection of C with the union of all algebraic subgroups of a certain codimension. As a corollary we obtain an explicit upper bound for the height of the rational points for some families of curves and an explicit version of the Manin-Dem’janenko method in products of elliptic curves