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 transfored into one-parameter Families C_s and D_s of cubic curves respectively. First we study the geometry of N_ε/G, ε=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(/-3) respectively. We show that C_s has the torsion group Z/3Z for a genric s∈Q and it also contains subfamilites 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 D_s has torsion Z/3Z+Z/3Z generically but also Z/3Z+Z/6Z for a subfamily which is para- Metrized by Q(/-3).
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机译：令N为具有3（3,4）尖点的六项式的模空间。商模空间N / G是一维的，由两个分量N_torus / G和N_gen / G组成。通过二次变换，它们分别转换为三次曲线的一参数族C_s和D_s。首先，我们研究了N_ε/ G的几何形状，ε= torus，gen及其椭圆形纤维的结构。然后我们分别研究了三次曲线C_s在Q上和D_s在Q（/-3）上的三次曲线的Mordell-Weil扭转群。我们证明C_s具有通用s∈Q的扭转群Z / 3Z，并且还包含与Kubert [Ku]给出的具有扭转群Z / 6Z，Z / 6Z + Z / 2Z的通用族相吻合的亚家族。 Z / 9Z或Z / 12Z。三次曲线D_s一般具有Z / 3Z + Z / 3Z扭转，但对于一个由Q（/-3）参数化的子族，Z_3Z + Z / 3Z也具有Z / 3Z + Z / 6Z。
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