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==Improvements to Schoof's Algorithm==
In the 1990s, [[Noam Elkies]], followed by [[A. O. L. Atkin]], devised improvements to Schoof's basic algorithm by restricting the set of primes <math>S = \{l_1, \ldots, l_s\}</math> considered before to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime <math>l</math> is called an Elkies prime if the characteristic equation: <math>\phi^2-t\phi+ q = 0</math> splits over <math>\mathbb{F}_l</math>, while an Atkin prime is a prime that is not an Elkies prime. Atkin showed how to combine information obtained from the Atkin primes with the information obtained from Elkies primes to produce an efficient algorithm, which came to be known as the [[Schoof-Elkies-Atkin algorithm]]. The first problem to address is to determine whether a given prime is Elkies or Atkin. In order to do so, we make use of modular polynomials, which come from the study of [[modular forms]] and an interpretation of elliptic curves over the complex numbers as lattices. Once we have determined which case we are in, instead of using Division Polynomials, we proceed by working modulo the modular polynomials <math>f_l</math> which have a lower degree than the corresponding division polynomial <math>\psi_l</math> (degree <math>O(l)</math> rather than <math>O(l^2)</math>). This results in a further reduction in the running time, giving us an algorithm more efficient than Schoof's, with complexity <math>O(\log^6 q)</math>.
==Implementations==
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