  Restricting ourselves to elliptic curves over $\mathbb{Q}$, we reformulate
the $p$-adic Beilinson conjecture due to Perrin-Riou, which is customized to
our computational approach. We then develop a new algorithm for numerical
verifications of the $p$-adic Beilinson conjecture, which is based on the
theory of rigid cohomology and $F$-isocrystals.
