If I am correct, an endomorphism is a homomorphism from one group to itself. In the case of an elliptic curve, we'd need a map $\phi: E \rightarrow E$. Ive seen in some places that you can get such a map by defining $\phi_n(P)=[n]P=P+P...+P$ (n times). This means if we represent P as $(\wp(t),\wp'(t))$, (over some arbitrary lattice) $\phi_n((\wp(t),\wp'(t)))=(\wp(nt),\wp'(nt))$.
Using this definition of a endomorphism found that $\phi_i$ is also a valid map if $E$ is parametrized by $(\wp(t,[1,i]),\wp'(t,[1,i]))$ since $\phi_i((\wp(t,[1,i]),\wp'(t,[1,i])))=(\wp(it,[1,i]),\wp'(it,[1,i]))=(-\wp(t,[1,i]),i\wp'(t,[1,i]))$.
That case was simple because the lattice $[1,i]$ is unchanged when divided by $i$. In a more general case where $E$ is parametrized by $(\wp(t,\Lambda),\wp'(t,\Lambda))$, how would one find other endomorphisms of E? Also, how does one prove these are homomorphisms? I havent been able to find any sort of reasoning for using such a map and think it would be useful to actually see why this is a homomorphism. Thanks for any help.