The order of a constant cycle curve \(C \subset X\) on a complex K3 surface, defined by Huybrechts, is a positive integer that measures the obstruction to decomposing the diagonal class \(\Delta _C\) in the Chow group \(\mathrm{CH}^2(X \times C)\) . In this paper, we compute the order of elliptic constant cycle curves that naturally arise on Kummer surfaces, by passing to the transcendental intermediate Jacobian \(J_{\textrm{tr}}^3(X \times C)\) . As a consequence, every \(n \in \mathbb {N}\) can be realized as the order of a constant cycle curve on a complex K3 surface.
