Summary: Fix an elliptic curve \(E\) over \(\mathbb{Q} \). An extremal prime for \(E\) is a prime \(p\) of good reduction such that the number of rational points on \(E\) modulo \(p\) is maximal or minimal in relation to the Hasse bound, i.e., \( a_p(E) = \pm \left [ 2 \sqrt{p} \right ]\). Assuming that all the symmetric power \(L\)-functions associated to \(E\) have analytic continuation for all \(s \in \mathbb{C}\) and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when \(E\) is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of J. Rouse and J. Thorner [Trans. Am. Math. Soc. 369, No. 5, 3575–3604 (2017; Zbl 1429.11086)], and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where \(a_p(E)\) is fixed because of the Sato-Tate distribution.