  We show that the elliptic genus of the higher rank E-strings can be computed
based solely on the genus 0 Gromov-Witten invariants of the corresponding
elliptic geometry. To set up our computation, we study the structure of the
topological string free energy on elliptically fibered Calabi-Yau manifolds
both in the unrefined and the refined case, determining the maximal amount of
the modular structure of the partition function that can be salvaged. In the
case of fibrations exhibiting only isolated fibral curves, we show that the
principal parts of the topological string partition function at given
base-wrapping can be computed from the knowledge of the genus 0 Gromov-Witten
invariants at this base-wrapping, and the partition function at lower
base-wrappings. For the class of geometries leading to the higher rank
E-strings, this leads to the result stated in the opening sentence.
