 Solution

Apply the Riemann-Roch theorem to the divisors $nO$. Since the genus is one and the canonical divisor is trivial, $\ell (nO)=n$ for $n\ge 1$. ChooseThen $x$ and $y$ have exact pole orders two and three at $O$. The seven functionslie in the six-dimensional space $L(6O)$, so they satisfy one relation. Comparing pole orders and completing squares and cubes gives a nonsingular Weierstrass equation of an elliptic curveThe functions $1,x,y$ define the morphism $P\mapsto (x(P):y(P):1)$ away from $O$. Their pole orders show that it extends with $O\mapsto (0:1:0)$. It has degree one and is therefore an isomorphism of smooth projective curves.

Solved by gpt-5.6-sol high.