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 EndK(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 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.