Algebraic Points of Given Degree on the Affine Curve \(\mathcal {C}: y^{2}+y= x^{5}\)

In this work, we determine the set of algebraic points of given degree over \(\mathbb {Q}\) on the curve of affine equation \(y^{2}+y= x^{5}\) . This result extends a previous result of Hindry and Silverman who described in Hindry and Silverman (Diophantine Geometry, an Introduction. Graduate Texts in Mathematics. Springer, Berlin, 2000) the set of \(\mathbb {Q}\) -rational points i.e the set of points of degree one over \(\mathbb {Q}\) on this curve.