In this paper, we present a study of elliptic curves, focusing on theirunderlying mathematical concepts, properties and applications in numbertheory. We begin by introducing elliptic curves and their unique features,discussing their algebraic structure, and exploring their group law, pro-viding examples and geometric interpretations. The core of our studyfocuses on the Elliptic Curve Method (ECM) for integer factorization.We present the motivation behind ECM and compare it to Pollard’s (p-1) method. A discussion on pseudocurves and the choice of an ellipticcurve and bound B is provided. We also address the differences betweenECM and Pollard’s (p-1) method and propose optimization techniques forECM, including the calculation of the least common multiple (LCM) ofthe first B integers using the Sieve of Eratosthenes.