The $5$–local homotopy of $eo_4$

Abstract

We compute the cohomology of a $5$–local analogue of the Weierstrass Hopf algebroid used to compute $tmf$–homology. We also compute the Adams–Novikov differentials for various stages, finding the homotopy, $V\left(0\right)$–homology and $V\left(1\right)$–homology of the putative spectrum $e{o}_4$. We also link this computation to the homotopy of the higher real $K$–theory spectrum $E{O}_4$.