Towards the construction of elliptic curves over Q with high rank and a point of order 5

By representing a genus one curve as a plane curve with five double points, we are able to construct a 3-parameter family of genus one curves over Q with Jacobians having a torsion subgroup isomorphic to Z(5). This leads, by specializing the parameters, to elliptic curves over Q of the Mordell-Weil group with high rank and with a torsion subgroup isomorphic to Z(5). We also show this family contains as a subfamily the principal homogeneous space parameterizing elliptic curves with a rational point of order 5, namely X-1 (5). We explicitly describe these families by equations in the Weierstrass form.