  This paper investigates the detection of the rank of elliptic curves with
ranks 0 and 1, employing a heuristic known as the Mestre-Nagao sum
  \[ S(B) = \frac{1}{\log{B}} \sum_{\substack{p<B \\ \textrm{good reduction}}}
\frac{a_p(E)\log{p}}{p}, \] where $a_p(E)$ is defined as $p + 1 -
\#E(\mathbb{F}_p)$ for an elliptic curve $E/\mathbb{Q}$ with good reduction at
prime $p$. This approach is inspired by the Birch and Swinnerton-Dyer
conjecture.
  Our observations reveal an oscillatory behavior in the sums, closely
associated with the recently discovered phenomena of murmurations of elliptic
curves. Surprisingly, this suggests that in some cases, opting for a smaller
value of $B$ yields a more accurate classification than choosing a larger one.
For instance, when considering elliptic curves with conductors within the range
of $[40\,000,45\,000]$, the rank classification based on $a_p$'s with $p < B =
3\,200$ produces better results compared to using $B = 50\,000$. This
phenomenon finds partial explanation in the recent work of Zubrilina.
