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In this paper we relate some classical normal forms for complex ellipticcurves in terms of 4-point sets in the Riemann sphere. Our main result is analternative proof that every elliptic curve is isomorphic as a Riemann surfaceto one in the Hesse normal form. In this setting, we give an alternative proofof the equivalence between the Edwards and the Jacobi normal forms. Also, wegive a geometric construction of the cross ratios for 4-point sets in generalposition.

Some remarks on the correspondence between elliptic curves and four points in the Riemann sphere