The Schoof-Elkies-Atkin (SEA) algorithm is currently the most efficient algorithm for counting the number of points of an elliptic curve defined over a finite field of large characteristic. The main idea of this algorithm is to use the relation between the order of the curve and the trace of the Frobenius endomorphism and then to compute this trace modulo small primes. Using the CRT and the Hasse-Weil bound leads to find the exact value of the trace. The implementation of SEA in PARI/GP is based on Reynald Lercier's thesis, published in 1997. Many improvements have been proposed since. In this talk, I will present two algorithms (respectively published by Gaudry and Morain and by Mihailescu, Morain and Schost) to compute the trace in the so-called Elkies case, their implementations in PARI and comparisons I made during my master's internship in the French Network and Information Security Agency.