  Franz Lemmermeyer's previous work laid the framework for a description of the
arithmetic of Pell conics, which is analogous to that of elliptic curves. He
describes a group law on conics and conjectures the existence of an analogous
Tate--Shafarevich group with order the squared ideals of the narrow class
group. In this article, we provide a cohomological definition of the
Tate--Shafarevich group and show that its order is as Lemmermeyer conjectured.
