An explicit proof of the weak finite basis theorem and applications to computing ranks of elliptic curves

Let E be an elliptic curve defined over the field Q of rational numbers, and let G be the group EQ of rational points of E . The classical proof of Mordell's Weak Finite Basis Theorem shows that G/2G is finite by embedding it in a certain finite group H whose genesis is algebraic number-theoretical. Assuming that G has trivial 2-torsion, we provide an explicit parametrization of H . This parametrization yields an upper bound for the rank of G as well as a heuristic algorithm to determine the exact rank. We offer some examples to illustrate the use and limitations of this approach.