Modular curves provide the first known examples of codes which are better than random ones. However, for an explicit construction of a code one needs a nonsingular model of the modular curve that initially is defined via a very singular planar model given by a modular equation. In this work we analyze the structure of its affine singularities in terms of class numbers of binary quadratic forms. In principle this allows to describe the space of regular differentials vanishing at a point needed for the Goppa construction.

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