(untitled) (Communications in Mathematical Research, 2020)
Isogenies occur throughout the theory of elliptic curves. Recently, the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols. Given two elliptic curves E_1,E_2 defined over a finite field k with the same trace, there is a nonconstant isogeny β from E_2 to E_1 defined over k. This study gives out the index of Hom_k(E_1,E_2)β as a nonzero left ideal in End_k(E_2) and figures out the correspondence between isogenies and kernel ideals. In addition, some results about the non-trivial minimal degree of isogenies between two elliptic curves are also provided.