  We study M-Theory solutions with $G$-flux on the Fermat sextic Calabi-Yau
fourfold, focussing on the relationship between the number of stabilized
complex structure moduli and the tadpole contribution of the flux. We use two
alternative approaches to define the fluxes: algebraic cycles and
(appropriately quantized) Griffiths residues. In both cases, we collect
evidence for the non-existence of solutions which stabilize all moduli and stay
within the tadpole bound
