Edwards curves and gaussian hypergeometric series

Let E be an elliptic curve described by either an Edwards model or a twisted Edwards model over F_p, namely, E is defined by one of the following equations x² + y² = a²(1 + x²y²), a⁵ - a ≠ 0 mod p, or, ax² + y² = 1 + dx²y², ad(a - d) ≢ 0 mod p, respectively.We express the number of rational points of E over F^p using the Gaussian hypergeometric series ₂F₁ (Formula Presenrted) where ϵ and φ are the trivial and quadratic characters over F_p respectively. This enables us to evaluate |E(F_p)| for some elliptic curves E, and prove the existence of isogenies between E and Legendre elliptic curves over F_p.