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?
Thank you for your help!