The longest common extension problem (LCE problem) is to construct a data structure for an input string $T$ of length $n$ that supports LCE$(i,j)$ queries. Such a query returns the length of the longest common prefix of the suffixes starting at positions $i$ and $j$ in $T$. This classic problem has a well-known solution that uses $O(n)$ space and $O(1)$ query time. In this paper we show that for any trade-off parameter $1 \leq τ\leq n$, the problem can be solved in $O(\frac{n}τ)$ space and $O(τ)$ query time. This significantly improves the previously best known time-space trade-offs, and almost matches the best known time-space product lower bound.