  We develop an algorithm to test whether a non-CM elliptic curve
$E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular
curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a
method for computing the image of the adelic Galois representation associated
to $E$. Running this algorithm on all elliptic curves presently in the
$L$-functions and Modular Forms Database and the Stein-Watkins Database gives
strong evidence for the conjecture that $E$ gives rise to an isolated point on
$X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or
$-162677523113838677$.
