Parameterization of algebraic points of a given degree on the curve of the affine equation \(y^3=x(x - 1)(x - 2)(x - 3)\) (Q611152)

\textit{E. F. Schaefer} [Math. Ann. 310, No. 3, 447--471 (1998; Zbl 0889.11021)] described the algebraic points of degree 2 and 3 on the curve \(C : y^3 = x(x-1)(x-2)(x-3)\) over \(\mathbb{Q}\). By using the computation of the Mordell-Weil group of the Jacobian of \(C\), the Abel-Jacobi theorem, and the study of linear systems on \(C\), the authors generalize the mentioned description to algebraic points of any given degree. The result is very explicit for degrees \(\leq 5\).