Let $E/ F_p$ be an elliptic curve. Let $G_A$, $G_{A'}$, $G_B$ and $G_{B'}$ be subgroups of $E(\overline{F_p})$. Suppose $E/G_A \simeq E/G_{A'}$ and $E/G_B \simeq E/G_{B'}$. When is $E/<G_A,G_B> \simeq E/<G_{A'},G_{B'}>$?
The above question was my attempt at finding necessary and sufficient conditions that an elliptic curve $E'$ (upto isomorphism) should satisfy so that the following diagram commutes: $\require{AMScd}$ \begin{CD} E @> \phi_A>> E/G_A\\ @V \phi_B V V @VV V\\ E/G_B @> > > E' \end{CD} I am assuming that all of these are cyclic isogenies and $(\deg \phi_A, \deg \phi_B)=1$. I wanted to say that $E'$ makes the diagram commute iff $(y-j(E')) | gcd(\Phi_{l_2}(j(E_A),y), \Phi_{l_1}(j(E_B),y))$, where $\Phi_i$ is the $i$-th modular polynomial and $l_1$ and $l_2$ are the degrees of the isogenies $\phi_A$ and $\phi_B$, resp. But this doesn't seem enough considering my original question.