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The basics of the theory of elliptic curves should be known to a large audience of mathematicians and computer scientists. In particular, everyone concerned with cryptography should know at least the fundamental aspects of this important topic. The purpose of the present textbook is to give an elementary introduction to elliptic curves. This branch of number theory is particularly accessible to computer-assisted calculations, and the authors make use of this feature by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH is applied on several occasions. However, the book can also be read by those not interested in the computational aspects. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated in its application. Hence, the authors made a careful selection of the topics treated. These topics include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The content is kept as elementary as possible, and therefore the book differs significantly from the numerous textbooks on elliptic curves nowadays available. The book is addressed to graduate students and researches in both mathematics and computer science. Table of Contents Preface Elliptic curves Elliptic curves over the complex numbers Elliptic curves over finite fields Elliptic curves over local fields The Mordell–Weil theorem and heights Torsion group The rank Basis Bibliography Index