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Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant$0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin\mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing arational $\mathbb{F}_{\!q}$-curve on the Kummer surface of the direct product$E_b \!\times\! E_b^\prime$, where $E_b^\prime$ is the quadratic$\mathbb{F}_{\!q}$-twist of $E_b$. More precisely, we propose to search such acurve among infinite order $\mathbb{F}_{\!q}$-sections of some elliptic surfaceof $j=0$, analyzing its Mordell--Weil group. Unfortunately, we prove that it isjust isomorphic to $\mathbb{Z}/3$.

Hashing to elliptic curves of $j=0$ and Mordell--Weil groups