Nice elliptic curves

Abstract

In this work, we study elliptic curves of the form $E_L: y^2 = x(x-1)(x-L)$, where $L^2-L+1 \in \left(\mathbb{Q}^{\times}\right)^2$ and $L \in \mathbb{Q} \setminus \{0,1\}$. We will show that for almost all quadratic extensions $K/\mathbb{Q}$ except one we have $E_L(K) \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.