A $1$-dimensional formal group over the prismatization of $\operatorname{Spf}\:\mathbb{Z}_p$

Let $\Sigma$ denote the prismatization of $\operatorname{Spf}\:\mathbb{Z}_p$. The multiplicative group over $\Sigma$ maps to the prismatization of $\mathbb{G}_m \times \operatorname{Spf}\:\mathbb{Z}_p$. We prove that the kernel of this map is the Cartier dual of some $1$-dimensional formal group over $\Sigma$. We obtain some results about this formal group (e.g., we describe its Lie algebra). We give a very explicit description of the pullback of the formal group to the quotient stack $Q/\mathbb{Z}^\times_p$, where $Q$ is the $q$-de Rham prism.