Algebraic points on the curve of affine equation \(y^2 =x(x-3)(x-4)(x-6)(x-7)\)

Abstract

In this work, we use the finiteness of the Mordell–Weil group of the Jacobian variety of the curve \(\mathcal {C}:y^2 =x(x-3)(x-4)(x-6)(x-7)\) and the Riemann Roch spaces to determine explicitly the set of algebraic points of given degree l over \(\mathbb {Q}\) on the curve \(\mathcal {C}\). The results obtained extend the work of Gordon and Grant, who determined the Mordell–Weil group \(J(\mathbb {Q})\) and the set of rational points on the same curve.