Update (Dec. 22): I have already solved this question with Magma.
Recently, I read a paper [1] and saw the following equation: $$u^3−2u^2−2v^3−20v^2+16v=0.$$ The author then got a Weierstrass equation $$Y^2=X^3-X^2-41X+441$$ through such transformation: $$X=\dfrac{-4u-3v+8}{v},$$ $$Y=-2\left(\dfrac{-u^3+4u^2-4u+2v^3+10v^2}{v^2}\right).$$ I know Nagell's algorithm (see, e.g., [2]) is a good approach to deal with such equation. But when I tried to transform it through Nagell's algorithm, I got a weird equation with large coefficients. My question is: how did the author get the transformation? Are there any tricks?
[1] P. Ingram, On kth-power numerical centers, C. R. Math. Acad. Sci. R. Can., 27 (2005), 105--110.
[2] R. J. Stroeker and B. M. M. de Weger, Solving elliptic Diophantine equations: the general cubic case, Acta Arith., 87 (1999), 339--365.