On isogeny classes of Edwards curves over finite fields

Original · EN

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 complete Edwards curve, and that an Edwards curve is isogenous to an original Edwards curve over if and only if its group order is divisible by 8 if q ≡ -1 4, and 16 if q ≡ 1 4. Furthermore, we give formulae for the proportion of d ∈ {0,1} for which the Edwards curve Ed is complete or original, relative to the total number of d in each isogeny class.