  We compute the mod $2$ homology of the spectrum $\mathrm{tmf}$ of topological
modular forms by proving a 2-local equivalence $\mathrm{tmf} \wedge DA(1)
\simeq \mathrm{tmf}_1(3) \simeq BP\left \langle 2\right\rangle$, where $DA(1)$
is an eight cell complex whose cohomology "doubles" the subalgebra
$\mathcal{A}(1)$ of the Steenrod algebra generated by $\mathrm{Sq}^1$ and
$\mathrm{Sq}^2$. To do so, we give, using the language of stacks, a modular
description of the elliptic homology of $DA(1)$ via level three structures. We
briefly discuss analogs at odd primes and recover the stack-theoretic
description of the Adams-Novikov spectral sequence for $\mathrm{tmf}$.
