Ahmadi, Omran
Granger, Robert
On isogeny classes of Edwards curves over finite fields
Journal of Number Theory
0022-314X
10.1016/j.jnt.2011.12.013
132
6
1337-1358
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 $\mathbb{F}_q$ 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 $d \in \mathbb{F}_q$ \ {0,1} for which the Edwards curve $E_d$ is complete or original, relative to the total number of $d$ in each isogeny class.
Edwards curves;
Legendre curves;
Isogeny classes;
Elsevier
2012
http://www.sciencedirect.com/science/article/pii/S0022314X1200025X;