The failure of GCH at a degree of supercompactness

We determine the large cardinal consistency strength of the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$. Indeed, we show that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that $2^\lambda \geq \theta$ is equiconsistent with the existence of a $\lambda$-supercompact cardinal that is also $\theta$-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.