# Finding a curve with maximal number of $\mathbb{F}_{q}$-points [duplicate]

Let $$q = p^{n}$$ and let $$E$$ be an elliptic curve. Hasse's bound from the Weil's conjecture tell us that $$|\sharp E(\mathbb{F}_{q}) - q - 1| \leq 2\sqrt{q}$$ for any $$q$$. Since this is a uniform bound over $$E$$, there should exists an elliptic curve $$E$$ that $$\sharp E(\mathbb{F}_{q})$$ takes a maximum value among all the elliptic curves (over $$\mathbb{Q})$$.
My question is: is there any algorithm to find such $$E$$ for any given $$q$$? It seems that the distribution for fixed $$p$$ and varying $$E$$ is something called 'vertical' Sato-Tate conjecture, and the above bound seems to be sharp.