How can you show that the complex number $(u,v)$, where $u,v \in \mathbb{C}$, satisfying the equation
$$P(u,v) = \sum_{i,j = 0}^{2} a_{ij} u^iv^j = 0$$
determines an elliptic curve.
Some Background-
I am reading I. M. Krichever's paper, “Baxter's equations and algebraic geometry”, in which he claims that the points $(u,v)$ form an elliptic curve in $\mathbb{C}$. As per my understanding, the form of elliptic curve in $\mathbb{C}$ is
$Y^2 = X^{3} + aX + b$
where discriminant is 0.