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.



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