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 a rational $\mathbb{F}_{\!q}$-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 a curve among infinite order $\mathbb{F}_{\!q}$-sections of some elliptic surface of $j=0$, analyzing its Mordell--Weil group. Unfortunately, we prove that it is just isomorphic to $\mathbb{Z}/3$.

中文翻译：

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散列到 $$j=0$$ 和 Mordell-Weil 群的椭圆曲线

考虑一条普通椭圆曲线 $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) 在有限域 $\mathbb{F}_{\!q}$ 上，使得 $ \sqrt[3]{b} \notin \mathbb{F}_{\!q}$。本文试图解决在直积$E_b \!\times\!的Kummer面上构造有理$\mathbb{F}_{\!q}$-曲线的问题。E_b^\prime$，其中 $E_b^\prime$ 是 $E_b$ 的二次 $\mathbb{F}_{\!q}$-twist。更准确地说，我们建议在 $j=0$ 的某个椭圆曲面的无限阶 $\mathbb{F}_{\!q}$-截面中搜索这样一条曲线，分析其 Mordell--Weil 群。不幸的是，我们证明它只是同构于 $\mathbb{Z}/3$。