= 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 $n\geq1$. Choose
$$
x\in L(2O)\setminus L(O),
\qquad y\in L(3O)\setminus L(2O).
$$
Then $x$ and $y$ have exact pole orders two and three at $O$. The seven functions
$$
1,x,y,x^2,xy,x^3,y^2
$$
lie 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 curve>
$$
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.
$$
The 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.
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