The Congruent Number Problem Part II: the Torus, the Elliptic Curve, and the Addition Law!

In the second in this series of videos, we discuss the equivalence of complex tori and projective complex elliptic curves via the Weierstrass phe-function. We then define the group addition law on an arbitrary elliptic curve analytically and geometrically.
Regarding the question posed at the end of the lecture; my answer was incomplete. Here is the correct answer: one can show using the Riemann-Roch theorem that every invertible sheaf of degree 1 on a genus 1 curve over a field K has one section up to scaling. This immediately gives a bijection between K-points on the curve and degree 1 invertible sheaves on the curve. Consider an elliptic curve, which is a genus 1 curve with a distinguished point. By twisting the above correspondence using the distinguished point, you end up with a bijection between degree 0 invertible sheaves and K-points on the elliptic curve. But the degree 0 sheaves form a group under the tensor product, and it is easy to see that associativity there transports naturally to the elliptic curve group law.

[@mathstrek]

To motivate the elliptic curve group law, you don't need the full power of schemes & sheaves. More simply, just look at the ideal class group of the coordinate ring of the elliptic curve. Recall that if A is an integral domain, the ideal class group of A is Cl(A) = (invertible fractional ideals of A) / (principal fractional ideals of A). Examples:
1) For A = Z[sqrt(-5)], the group Cl(A) is finite of order 2; in fact more generally Cl(A) is finite for any number ring A.
2) When A = C[E], where E is an elliptic curve over the complex field C, the class group Cl(A) is exactly {points of E} + {1}. This can be proven via elementary methods.

[@fotisp6293]

Hey, could you maybe give me a hint/walkthrough on how one would go on to prove that the lattice for the curve y^2=x^3-n^x is a multiple of the Gaussian integer lattice?
I assume the first step would be to bring the curve to the form y^2=4t^3-g_2*t-g_3 through the substitution x=4^(1/3)t, but I dont know how to continue from there.
Thank you for your time

[@Entropize1]

I would recommend looking at the specific formulas for g_2, g_3 since you know what they must equal.