The existence of a modular invariant genus two partition function implies infinitely many relations among the structure constants of a chiral self-dual conformal field theory. All of these relations can be shown to be a consequence of the associativity of the OPE, as well as the modular covariance properties of the torus one-point functions. Using these techniques we prove that for the proposed extremal conformal field theories at c=24k a consistent genus two vacuum amplitude exists for all k, but that this does not actually check the consistency of these theories beyond what is already testable at genus one.