We count the number of isogeny classes of Edwards curves over finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a { \em complete} Edwards curve, and that an Edwards curve is isogenous to an { \em original} Edwards curve over $\F_q$ if and only if its group order is divisible by 8 if $q \equiv -1 \pmod{ 4}$, and 16 if $q \equiv 1 \pmod{ 4}$. Furthermore, we give formulae for the proportion of $d \in \F_q \setminus \{ 0,1\}$ for which the Edwards curve $E_d$ is complete or original, relative to the total number of $d$ in each isogeny class.
Categories: math.NT cs.CR