  We compute the 5-local cohomology of a 5-local analogue of the Weierstrass
Hopf algebroid used to compute $tmf$ homology. We compute the Adams-Novikov
differentials in the cohomology, giving the homotopy, V(0)-homology, and
V(1)-homology of the putative spectrum $eo_4$. We also link this computation to
the homotopy of the higher real $K$-theory spectrum $EO_4$.
