This is problem 8.2 from Fulton's Algebraic Curves. Let $X = C = V(Y^2 Z - X(X-Z)(X- \lambda Z)) \in \mathbb{P^2}, \lambda \neq 0,1$. Compute div$(x)$ and div$(y)$, where $x = X/Z$, $y = Y/Z$.
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Related: https://math.stackexchange.com/questions/511950/calculating-the-divisors-of-the-coordinate-functions-on-an-elliptic-curve – Benjamin Jul 17 '21 at 20:11