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In 2009, Banks and Shparlinski proved an averaged version of this result over families of elliptic curves. In this article, we give a more explicit analysis of these densities. In particular, we show that, for Serre curves, the density of primes $p$ for which $m \\mid \\#E_p(\\mathbb{F}_p)$ is approximately $1/\\varphi(m)$, and is always greater than $1/m$ for every $m \\geq 2$. 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