  Let $\mathcal N$ be the moduli space of sextics with 3 (3,4)-cusps. The
quotient moduli space ${\mathcal N}/G$ is one-dimensional and consists of two
components, ${\mathcal N}_{torus}/G$ and ${\mathcal N}_{gen}/G$. 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 $\bfQ$ and $D_s$ over $\bfQ(\sqrt{-3})$ respectively.
We show that $C_{s}$ has the torsion group $\bf Z/3\bf Z$ 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
  $\bf Z/6\bf Z$, $\bf Z/6\bf Z+\bf Z/2\bf Z$, $\bf Z/9\bf Z$ or $\bf Z/12\bf
Z$. The cubic curves $D_s$ has torsion $\bf Z/3\bf Z+\bf Z/3\bf Z$ generically
but also $\bf Z/3\bf Z+\bf Z/6\bf Z$ for a subfamily which is parametrized by $
\bf Q(\sqrt{-3}) $.
