Change of variables for elliptic curve

Question

Say I have an elliptic curve $y^{2} + 5xy + y = x^{3}$ with the $(0, 0)$ being a rational 3-torsion point at $(x, y) = (0, 0)$. What change of variables would I need to get it into the form $y^{2} = x^{3} + a(x - b)^{2}$?

Answer 1

Complete the square for the left hand side: $$y^2+5xy+y = y^2+(5x+1)y = \left(y+\frac{5}{2}x+\frac{1}{2}\right)^2-\left(\frac{5}{2}x+\frac{1}{2}\right)^2.$$ Putting $Y := y+\frac{5}{2}x+\frac{1}{2}$, your equation then becomes $$Y^2 = x^3 + \left(\frac{5}{2}\right)^2\left(x+\frac{1}{5}\right)^2.$$