On arithmetic progressions on Edwards curves

Let m be a positive integer and a,q two rational numbers. Denote by AP_m(a,q) the set of rational numbers d such that a,a+q,...,a+(m-1)q form an arithmetic progression in the Edwards curve E_d:x^2+y^2=1+d x^2 y^2. We study the set AP_m(a,q) and we parametrize it by the rational points of an algebraic curve.