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Classification of AS-regular algebras is one of the main interests innoncommutative algebraic geometry. We say that a $3$-dimensional quadraticAS-regular algebra is of Type EC if its point scheme is an elliptic curve in$\mathbb{P}^{2}$. In this paper, we give a complete list of geometric pairs anda complete list of twisted superpotentials corresponding to such algebras. Asan application, we show that there are only two exceptions up to isomorphismamong all $3$-dimensional quadratic AS-regular algebras which cannot be writtenas a twist of a Calabi-Yau AS-regular algebra by a graded algebra automorphism.

A complete classification of $3$-dimensional quadratic AS-regular algebras of Type EC