A question about points on an elliptic curve with prime denominator

Abstract

Let E be an elliptic curve defined by a Weierstrass equation with integer coefficients. Any rational point on E other than the identity is of the form $(x(P)/z(P{)}^2,y(P)/z(P{)}^3)$ where $x(P),y(P)\in \mathbb{Z}$ and $z(P)\in \mathbb{N}$ and in addition both $\gcd (x(P),z(P))=1$ and $\gcd (y(P),z(P))=1$ hold. The question is: how often is $z(P)$ a prime number?

arXiv comment3 pages. This is a slightly edited version of an open question I submitted to the problem session of the JA2023 conference, where it was communicated by Michel Waldschmidt