Distribution of elliptic twin primes in isogeny and isomorphism classes

Motivated by a recent application to hash functions suggested by O. Chevassut, P.-A. Fouque, P. Gaudry and D. Pointcheval, we study the frequency with which groups of points on an elliptic curve over a finite field and its quadratic twist are both of prime order. We obtain a heuristic asymptotic formula for the number of isogeny and isomorphism classes of such curves and present some supporting computational results.