Given ordinary elliptic curves *E* and *E*' over a finite field
,
this paper
discusses the problem of determining for which
it holds that
.

**Justin T Miller**

are precisely those with order seven. They all have the same group structure over , namely , but over , and . Here ``smallest'' refers to the order of a field for which there are two elliptic curves

the group is cyclic, and when , with

where

Let

Then is a basis for as a -module, and where

If then and , so

and,

Theorem 3 provides a way to answer the question above for all

This theorem can be refined by replacing

have and as their endomorphism rings, respectively, but for all . If, however, for some