  Let $E$ be an elliptic curve described by either an Edwards model or a
twisted Edwards model over $\mathbb{F}_p$, namely, $E$ is defined by one of the
following equations $x^2+y^2=a^2(1+x^2y^2),\, a^5-a\not\equiv 0$ mod $p$, or,
$ax^2+y^2=1+dx^2y^2,\,ad(a-d)\not\equiv0$ mod $p$, respectively. We express the
number of rational points of $E$ over $\mathbb{F}_p$ using the Gaussian
hypergeometric series $\displaystyle {_2F_1}\left(\begin{matrix}
  \phi&\phi
  {} & \epsilon
  \end{matrix}\Big| x\right)$ where $\epsilon$ and $\phi$ are the trivial and
quadratic characters over $\mathbb{F}_p$ respectively. This enables us to
evaluate $|E(\mathbb{F}_p)|$ for some elliptic curves $E$, and prove the
existence of isogenies between $E$ and Legendre elliptic curves over
$\mathbb{F}_p$.
