Elliptic Curves from Sextics

Let $\mathcal{N}$ be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space $\mathcal{N}\mathrm{/}G$ is one-dimensional and consists of two components, ${\mathcal{N}}_{torus}\mathrm{/}G$ and ${\mathcal{N}}_{gen}\mathrm{/}G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. We study the Mordell-Weil torsion groups of cubic curves $C_s$ over $\text{bfQ}$ and $D_s$ over $\text{bfQ}(\sqrt{-3})$ respectively. We show that $C_s$ has the torsion group $\mathbf{Z}\mathbf{/}\mathbf{3}\mathbf{Z}$ for a generic $s\in \bf Q$ and it also contains subfamilies which coincide with the universal families given by Kubert with the torsion groups $\mathbf{Z}\mathbf{/}\mathbf{6}\mathbf{Z}$, $\mathbf{Z}\mathbf{/}\mathbf{6}\mathbf{Z}+\mathbf{Z}\mathbf{/}\mathbf{2}\mathbf{Z}$, $\mathbf{Z}\mathbf{/}\mathbf{9}\mathbf{Z}$ or $\bf Z/12\bf Z$\mathrm{.}Thecubiccurves$D_s$hastorsion$\bf Z/3\bf Z+\bf Z/3\bf Z$ generically but also $\mathbf{Z}\mathbf{/}\mathbf{3}\mathbf{Z}+\mathbf{Z}\mathbf{/}\mathbf{6}\mathbf{Z}$ for a subfamily which is parametrized by $\mathbf{Q}(\sqrt{-\mathbf{3}})$.