Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks

The theory of elliptic curves is one of the popular topics of recent times with its unsolved problems and interesting conjectures. In 1922, Mordell proved that the group of Q-rational points on an elliptic curve is finitely generated. However, the rank of this group, signifying the number of independent generators, can be arbitrarily high for certain curves, a fact yet to be definitively proven. This study leverages the computer algebra system Magma to investigate curves with potentially high ranks using a technique developed by Mestre.