# Finding preimage of point for isogenies between elliptic curves

Let's say one has an isogeny $$\alpha:E_1\to E_2$$ between two elliptic curves, and that $$\ker\alpha$$ is known. If there is a point $$S_2\in E_2$$, is there an efficient way to find its preimage $$S_1\in E_1$$ (where $$\alpha(S_1)=S_2$$) or any multiple of the preimage, on $$E_1$$?

I know that one possibility would be to compute the map of $$S_2$$ using the dual isogeny, i.e. that $$\hat\alpha(S_2)=[\deg\alpha]S_1$$, but then one would first need to generate and compute the dual isogeny $$\hat\alpha$$. I was wondering if there is a faster way, e.g. by analysing the action of $$\alpha$$ on different points or something I haven't thought of yet?