The rank of a CM elliptic curve and a recurrence formula

Let $p$ be a prime number and $E_p$ denote the elliptic curve $y^2=x^3+px$. It is known that for $p$ which is congruent to $1,9$ modulo $16$, the rank of $E_p$ over $\mathbb{Q}$ is equal to $0,2$. Under the condition that the Birch and Swinnerton-Dyer conjecture is true, we give a necessary and sufficient condition that the rank is $2$ in terms of the constant term of some polynomial that is defined by a recurrence formula.