By Hasse's theorem we know that range of the group order of the elliptic curve. And similarly, there exist a theorem on the admissible order of elliptic curves. Suppose by the theorem on the admissible order of elliptic curve we know there exist a curve of desired order then how to determine the curve parameters. For example in case of simplified/short Weierstrass equation the value of the parameters $a,b$.
My question: is there a theorem to determine the curve parameters based on the group order? Does the form of the curve matter? The different forms of the curve are Weierstrass, Montgomery, and Edwards.