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Here is a calculation regarding the $2$-torsion points of the elliptic curve $y^2=x^3+1$ which looks really miraculous to me (the motivation comes at the end). Take a point of $y^2=x^3+1$ and ...

I'm reading this post by Charles Siegel on Monodromy Representations and there is a construction in example a not unterstand. We look at the family $y^2z=x(x-z)(x-\lambda z)$ of projective elliptic ...