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Abstract. We completely solve diophantine equations of the form Y 2 = X 3 ± p k X, where k is a positive integer, using a reduction to some quartic elliptic equations, which can be solved with well known methods.

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We obtain a quantitative version of the classical Chevalley-Weil theorem for curves. Let φ :C → C be an unramified morphism of non-singular plane projective curves defined over a number field K. We calculate an effective upper bound for the norm of the relative discriminant of the number field K(Q) over K for any point P ∈ C(K) and Q ∈ φ −1 (P ).