 
Summary: On the probability of having rational isogenies
Jeffrey D. Achter and Daniel Sadornil
Abstract. We calculate the chance that an elliptic curve over a finite field has
a specified number of isogenies which emanate from it. We give a partial
answer for abelian varieties of arbitrary dimension.
Mathematics Subject Classification (2000). 11G20; 14K02.
Keywords. elliptic curve, isogeny, abelian variety, finite field.
1. Introduction
Suppose E and E are elliptic curves over a field K. A (Krational) isogeny is a
morphism : E E , defined over K, which takes the identity element of E to
that of E ; such a morphism is necessarily a group homomorphism. Let (E, , K)
be the number of Krational isogenies (up to isomorphism) of degree which em
anate from E. In this paper we analyze the distribution of (E, , Fq) as E ranges
over all elliptic curves over the finite field Fq.
Isogenies of prime order have algorithmic significance. For example, much
information about the trace of Frobenius tE of an elliptic curve E/Fq is encoded in
(E, , Fq). Indeed, let mE = t2
E  4q, and suppose that E is ordinary with j(E)
{0, 1728}, so that Aut¯Fq
(E) = {±1}. If mE
