On elliptic modular foliations, II

We give an example of a one-dimensional foliation $\mathcal{F}$ of degree two in a Zariski open set of a four-dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves $X_0(d)$, $d\in\mathbb{N}$, minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief was that such formulas do not exist and the emphasis in the literature has been on introducing faster algorithms to compute equations for small values of $d$.