Some context
It can be proved that for the elliptic curve given by $E :y^2 = x^3-x$, the number of points of $E$ in $\Bbb P^2(\Bbb F_p)$, denoted by $|E(\Bbb F_p)|$, is $p+1$ if and only if $p \equiv 3 \pmod 4$ (where $p>3$ is a prime number). In other words, the set $$S(E) := \{p \text{ prime numbers } \;:\; |E(\Bbb F_p)|=p+1\}$$ is equal to the set of the prime numbers of the form $P(n)=3n+4$.
However, I was wondering what happens for a different elliptic curve, namely $$E : y^2=x^3+5x+1.$$ The first primes $p$ such that $|E(\Bbb F_p)|=p+1$ are $5,131,193,521,1987,2089,2161,2503,2803,5189,6737,…$ and OEIS didn't give anything. Apparently, we can't find any "easy" congruence modulo some integer to list such primes, i.e. I think that these primes can't be represented as $an+b$ for fixed $a,b$, as $n≥N$ goes through $\Bbb N$ (for $N$ large enough). Whence the following question:
Question
Is there a polynomial $P \in \Bbb Z[X]$ with positive coefficients and a positive integer $N \in \Bbb N$, such that $$S(E) \cap \Bbb P_{≥N} = \{P(k) \mid k \in \Bbb N \} \cap \Bbb P_{≥N} $$ where $\Bbb P_{≥N}$ denotes the set of prime numbers greater that $N$ and $E : y^2=x^3+5x+1$ ?
In the previous example, we had $P(X)=3X+4$ and $N=5$. I am aware that this may be a difficult question, so I would be also interested in references about "polynomial growth" of the primes such that $a_p:=p+1-|E(\Bbb F_p)|=0$ (supersingular primes ?) for a given elliptic curve.
Thank you very much!