Arithmetic complexity revisited

The arithmetic complexity counts the number of algebraically independent entries in periodic continued fraction $\theta=[b_1,\dots, b_N, \overline{a_1,\dots,a_k}]$. If $\mathscr{A}_{\theta}$ is a noncommutative torus corresponding to rational elliptic curve $\mathscr{E}(K)$, then rank $\mathscr{E}(K)$ given by simple formula $r(\mathscr{E}(K))= c(\mathscr{A}_{\theta})-1$, where $c(\mathscr{A}_{\theta})$ $\theta$. We prove that equal dimension Brock-Elkies-Jordan variety $\theta$ introduced [1]. Following Zagier and Lemmermeyer, we evaluate Shafarevich-Tate group $\mathscr{E}(K)$.