Algebraic Points of Given Degree on the Curve of Affine Equation: \( {y^{2} =x^{5}+6912}\)

We explicitly define the set of algebraic points of given degree on the curve of affine equation \(y^{2}=x^{5}+6912 \) . This curve was studied by Nils Bruin in (On powers as sums of two cubes, Algorithmic Number Theory, Tome 21, 4th International Symposium, ANTS-IV Leiden, The Netherlands, July 2–7, 169–184, 2000), which shows that the Mordell-Weil group of the jacobian is finite. We use a basis of the Mordell-Weil group and the Abel-Jacobi theorem to describe the set of algebraic points of given degree on the same curve.