Let $O = [0:1:0]$ be the point at infinity, and define $f:\mathbb{R}\to\mathbb{R}$ by
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\begin{align*}
f(x)=x^3+ax+b.
\end{align*}
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If $f$ has exactly one real root, then $E(\mathbb{R})$ is connected. If $f$ has three distinct real roots, then $E(\mathbb{R})$ has exactly two connected components: one bounded affine oval and one component containing $O$.