Elliptic Curves over the Rationals

This chapter covers prerequisites on elliptic curves over \(\mathbb {Q}\) needed later in the book. The question of possible torsion groups of elliptic curves over \(\mathbb {Q}\) is discussed, with particular attention to those groups that can appear for curves induced by Diophantine triples. Methods for computing the rank and constructing elliptic curves with high rank are presented. The functions related to elliptic curves, available in the software package PARI/GP, are described.