The Adams–Novikov E2E2-term for Behrens' spectrum Q(2)Q(2) at the prime 3

We compute the Adams–Novikov E2E2-term of a spectrum Q(2)Q(2) constructed by M. Behrens. The homotopy groups of Q(2)Q(2) are closely tied to the 3-primary stable homotopy groups of spheres; in particular, they are conjectured to detect the homotopy beta family of Greek letter elements at the prime 3. Our computation leverages techniques used by Behrens to compute the rational homotopy of Q(2)Q(2), and leads to a conjecture that the Adams–Novikov E2E2-term for Q(2)Q(2) detects the algebraic beta family in the BP  -based Adams–Novikov E2E2-term for the 3-local sphere.