Some remarks on a theorem of Parent and generalizing Ogg's conjecture

We carry out an Eisenstein prime descent to prove the finiteness of the Mordell-Weil group of the Eisenstein quotients of J(1) (p) for certain values of p that are relevant to torsion in elliptic curves over cubic fields. We then use this to recover some results of Parent. Our methods suggest a possible generalization of Ogg's conjecture for torsion in elliptic curves over number fields.