Elliptic curves play an important role in number theory and cryptography. This report explores essential aspects of elliptic curves, such as their group structure and their torsion subgroup and isogenies - with particular emphasis on the Frobenius map. Special focus is given to Hasse's bound and division polynomials - both are an essential foundation for the study of René Schoof's algorithm described in [Sch85]. This algorithm, published in 1985, allows the computation of the number of points on an elliptic curve defined over a finite field with a significant time saving to previous approaches. This work provides a detailed analysis of this algorithm: we expand key steps which were only briefly mentioned, and even correct minor mistakes in the original document. To enhance understanding, we complement our report with detailed examples and SageMath-generated illustrations for many of the concepts covered
