On isogeny classes of Edwards curves over finite fields

We count the number of isogeny classes of Edwards curves over odd characteristic finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a complete Edwards curve, and that an Edwards curve is isogenous to an original Edwards curve over Fq if and only if its group order is divisible by 8 if q≡−1 , and 16 if q≡1 . Furthermore, we give formulae for the proportion of d∈Fq∖{0,1} for which the Edwards curve Ed is complete or original, relative to the total number of d in each isogeny class.