  We construct $G_4$ fluxes that stabilize all of the 426 complex structure
moduli of the sextic Calabi-Yau fourfold at the Fermat point. Studying flux
stabilization usually requires solving Picard-Fuchs equations, which becomes
unfeasible for models with many moduli. Here, we instead start by considering a
specific point in the complex structure moduli space, and look for a flux that
fixes us there. We show how to construct such fluxes by using algebraic cycles
and analyze flat directions. This is discussed in detail for the sextic
Calabi-Yau fourfold at the Fermat point, and we observe that there appears to
be tension between M2-tadpole cancellation and the requirement of stabilizing
all moduli. Finally, we apply our results to show that even though symmetric
fluxes allow to automatically solve most of the F-term equations, they
typically lead to flat directions.
