  We study the Jacobian scheme of a plane algebraic curve at an ordinary
singularity, characterizing it through a geometric property. We compute the
Tjurina number for a family of curves at an ordinary singularity showing that
it reaches the minimum possible value, using very elementary methods,
essentially Gr\"obner basis. We give an algorithm that gives the analytic type
of a double point using the algebraic version of the Mather-Yau Theorem.
