Is the group action on an elliptic curve reflecting the fact that it is a space with with transfers?

Through the recognition principle

$$Mon_{\mathbb{E}_\infty}(Spc)^{gp}\simeq Sp_{\geq0}\subset Sp=\:cohomology \:theories$$

there must be a cohomology theory for every elliptic curve. Is this elliptic cohomology developed by Lurie?