Elliptic curves from sextics

Let N be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space N/G is one-dimensional and consists of two components, N-torus/G and N-gen/G. By quadratic transformations, they are transformed into one-parameter families C-s and D-s of cubic curves respectively. First we study the geometry of N-epsilon/G, epsilon = torus, gen and their structure of elliptic fibration. Then we study the Mordell-Weil torsion groups of cubic curves C-s over Q and D-s over Q(root-3) respectively. We show that C-s has the torsion group Z/3Z for a generic s is an element of Q and it also contains subfamilies which coincide with the universal families given by Kubert [Ku] with the torsion groups Z/6Z, Z/6Z + Z/2Z, Z/9Z, or Z/12Z. The cubic curves Lis has torsion Z/3Z + Z/3Z generically but also Z/3Z + Z/6Z for a subfamily which is parametrized by Q(root-3).