  We study the differential equations governing mirror symmetry of elliptic
curves, and obtain a characterization of the ODEs which give rise to the
integral ${\bf q}$-expansion of mirror maps. Through theta function
representation of the defining equation, we express the mirror correspondence
in terms of theta constants. By investigating the elliptic curves in
$X_9$-family, the identification of the Landau-Ginzburg potential with the
spectral curve of Ising model is obtained. Through the Jacobi elliptic function
parametrization of Boltzmann weights in the statistical model, an exact Jacobi
form-like formula of mirror map is described .
