Computing the geometric endomorphism ring of a genus 2 Jacobian

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute ${\mathrm{End}}_{\bar{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\mathrm{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.