Elliptic curves over finite fields with Fibonacci numbers of points

For a prime power q and an elliptic curve E over F-q having q +1- a points, where a is an element of [-2 root q, 2 root q] let {#E-m}, m >= 1 be the sequence of numbers whose mth term is the number of points of E over F-q(m). In this paper, we determine all instances when #({#E-m}(m >= 1) boolean AND {F-n}(n >= 1)) >= 2, where {F-n}(n >= 1) is the sequence of Fibonacci numbers. That is, we determine all six-tuples (a, q, m(1), m(2), n(1), n(2)) such that #E = q +1- a, #E-m1 = F-n1 and #E-m2 = F-n2.