The 5-local homotopy of $eo_4$

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 $EO_4$.

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