Computing nonsurjective primes associated to Galois representations of genus $2$ curves

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre’s open image theorem for abelian surfaces asserts that there are only finitely many primes $\mathrm{\ell }$ for which the Galois action on $\mathrm{\ell }$-torsion points of $A$ is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell’s classification of maximal subgroups of $\mathrm{P}\mathrm{S}{\mathrm{p}}_4({\mathbb{F}}_{\mathrm{\ell }})$, sampling of the characteristic polynomials of Frobenius, and the Khare–Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus~$2$ curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve.