  We study an interplay between operator algebras and geometry of rational
elliptic curves. Namely, let $\mathcal{O}_B$ be the Cuntz-Krieger algebra given
by square matrix $B=(b-1, ~1, ~b-2, ~1)$, where $b$ is an integer greater or
equal to two. It is proved, that there exists a dense self-adjoint sub-algebra
of $\mathcal{O}_B$, which is isomorphic (modulo an ideal) to a twisted
homogeneous coordinate ring of the rational elliptic curve $\mathcal{E}({\Bbb
Q})=\{(x,y,z) \in {\Bbb P}^2({\Bbb C}) ~|~ y^2z=x(x-z)(x-{b-2\over b+2}z)\}$.
