An elliptic curve E can be immersed in \({\textbf{P}}^{N-1}\) as a curve of degree N by means of the linear system of |NO|, where O is the origin of E. Well-known classical results going back to Bianchi and Klein say that if N is odd, this immersion is uniquely determined by specifying a full-level N structure. In this paper we show that if N is even, uniqueness of immersion is ensured by specifying a level structure associated with a certain congruence subgroup between \(\Gamma (N)\) and \(\Gamma (2N)\) . Moreover, we construct, over the complex number field, an immersion by means of suitably chosen theta functions, and write down the quadratic equations satisfied by them.
