Let \(E\) be an elliptic curve, \(j\) an embedding of \(E\) in \(\mathbb{P}^ 3\) and \(\tau \in E\) not of order 4. The authors show that many algebraic properties of the Sklyanin algebra \(A\) associated to this data, first defined in [E. K. Sklyanin, Funct. Anal. Appl. 16, 263-270 (1983); translation from Funkts. Anal. Prilozh. 16, 27-34 (1982; Zbl 0513.58028)], have geometric interpretations in terms of \(E,j, \tau\). They first prove the equivalence between the original definition and a geometric one given in the present paper. A point, resp. line, plane, module is a graded module whose Hilbert series coincides with the Hilbert series of a polynomial ring in 1, resp. 2, 3, variables. One of the main results of this article is the characterization of the Cohen-Macaulay (graded) modules over \(A\) of multiplicity 1 and Gelfand-Kirillov dimension 1,2,3: they are precisely shifts of point, resp. line, plane, modules. It is also proved that line modules are in bijective correspondence with lines in \(\mathbb{P}^ 3\) secant to \(E\). (Analogously, it was proved in [the second author and J. T. Stafford [Compos. Math. 83, 259-289 (1992; Zbl 0758.16001)] that point modules are in bijective correspondence with points of \(E\) plus 4 more points; these 4 points also play a role in the new definition of \(A\) and are, in fact, the only points which lie on infinitely many secants of \(E)\).