  Let $F\in\mathbb{Z}[x,y]$ be some polynomial of degree 2. In this paper we
find the asymptotic behaviour of the least common multiple of the values of $F$
up to $N$. More precisely, we consider $\psi_F(N) =
\log\left(\text{LCM}_{0<F(x,y)\leq N}\left\lbrace F(x,y)\right\rbrace\right)$
as $N$ tends to infinity. It turns out that there are 4 different possible
asymptotic behaviours depending on $F$. For a generic $F$, we show that the
function $\psi_F(N)$ has order of magnitude $\frac{N\log\log N}{\sqrt{\log
N}}$. We also show that this is the expected order of magnitude according to a
suitable random model. However, special polynomials $F$ can have different
behaviours, which sometimes deviate from the random model. We give a complete
description of the order of magnitude of these possible behaviours, and when
each one occurs.
