PROBLEM: Put the Legendre equation $y^2 = x(x − 1)(x − λ)$ into Weierstrass form and use this to show that the j-invariant is j = $2^8\frac{(λ2 − λ + 1)^3}{λ^2(λ − 1)^2}$ .
Recall: Weierstrass equation form: E: y^2 = x^3 + Ax +B and
J(E) = 1728$\frac{4A^3}{4A^3+27B^2}$
Attempt: $y^2 = x^3 + (-\lambda-1)x^2 + \lambda x$
Now I'm having trouble transforming this into Weierstass form. What would be the way to go for transforming this equation? $x^2 = x_1$ would not work. Idk..
In general, if a curve is given by $Y^2=X^3+AX^2+BX+C$, a change of variables $Y=y$ and $X=x-A/3$ will provide a model for the curve of the form $y^2=x^3+A'x+B'$.
The reason is that $$X^3+AX^2+\cdots = (x-A/3)^3 + A(x-A/3)^2+\cdots$$ and the coefficient in $x^2$ is given by $3(-A/3)+A=0$.