On isogeny classes of Edwards curves over finite fields

Abstract

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 F q if and only if its group order is divisible by 8 if q ≡ − 1 ( mod 4 ) , and 16 if q ≡ 1 ( mod 4 ) . Furthermore, we give formulae for the proportion of d ∈ F q ∖ { 0 , 1 } for which the Edwards curve E d is complete or original, relative to the total number of d in each isogeny class.