High Quality Content by WIKIPEDIA articles! The Elkies-Atkin extension to Schoof's algorithm works by restricting the set of primes S = {l_1, ldots, l_s} considered to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime l is called an Elkies prime if the characteristic equation: ?2 ? t? + q = 0 splits over mathbb{F}_l, 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.

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