This paper studies point modules and line modules over the algebra defined by E.K. Sklyanin in [17]. It was proved in [21] that the point modules are in bijection with the points of an elliptic curve $E$ in $\mathbb {P}^3$ together with four other points. Here it is proved that the line modules are in bijection with the lines in $\mathbb {P}^3$ which are secant lines to $E$. The point and line modules are precisely the Cohen-Macaulay modules of multiplicity $1$, and Gelfand-Kirillov dimension $1$ and $2$ respectively. Further algebraic properties of these modules are shown to be consequences and analogues of the geometric properties of the elliptic curve and the four points. For example, if two lines lie on a smooth quadric containing $E$, and they do not intersect, then the two corresponding line modules have the same annihilator. It is also shown that the Sklyanin algebra may be defined in terms of the bilinear forms vanishing on a certain subvariety of $\mathbb {P}^3 \times \mathbb {P}^3$.