In a joint work with P. Bayer [Ba-La 91], we verify Serre's conjecture (3.2.4$_?$) (in Duke J.) for the Galois representation defined by the $p$-torsion points of $p$-vertical Weil curves. In this paper our purpose is to emphasize the difference between the $p$-vertical and the $p$-horizontal cases in order to check Serre's conjecture. Several numerical examples, collected by computer calculations, lead us to give a conjecture which implies (3.2.4$_?$) for the horizontal case.