Computation of the Mordell-Weil Group

Abstract

If G is an abelian group (written additively), the elements g ₁,..., g _r of G are called independent if

$$m_1 g_1 + \cdots m_r g_r = 0\;\left( {m_j \varepsilon \mathbb{Z}} \right)$$

is possible only with m ₁ = ⋯ = m _r = 0. Thus if one of g ₁,..., g _r is of finite order, g ₁,..., g _r cannot be independent. For any elliptic curve E defined over ℚ the group E(ℚ) of rational points on E is finitely generated. The (Mordell-Weil) rank r _ℚ(E) of E is defined to be the maximum number of independent elements in E(ℚ). In particular, r _ℚ(E) = 0 if and only if E(ℚ) is finite (consisting of points of finite order).