Towards a classification of isolated $j$-invariants

We develop an algorithm to test whether a non-CM elliptic curve $E\mathrm{/}\mathbb{Q}$ 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 $j(E)=-140625\mathrm{/}8,-9317,$ $351\mathrm{/}4$, or $-162677523113838677$.