  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 $E_L(K) \cong \mathbb{Z}/2\mathbb{Z} \oplus
\mathbb{Z}/2\mathbb{Z}$.
