The Cox-Zucker Machine is an algorithm created by David A. Cox and Steven Zucker. This algorithm determines if a given set of sections provides a basis (up to torsion) for the Mordell–Weil group of an elliptic surface $E \to S$ where $S$ is isomorphic to the projective line.
The algorithm was first published in Intersection numbers of sections of elliptic surfaces (1979), by Cox and Zucker, and it was later named the "Cox–Zucker machine" by Charles F. Schwartz in A Mordell-Weil group of rank 8, and a subgroup of finite index (1984).
And I did check the paper by Charles Schwartz, and indeed:
We will find, for a specific equation of this form, $$y^2 = 4(x^3 - u^4x + 1),$$ 8 solutions that generate a subgroup of index 4 in the Mordell-Weil group of the fibration given by this equation. We do this using the Cox-Zucker Machine. We then use this result to draw certain conclusions concerning the general case, and to make certain conjectures.
§ 1. The algorithm of Cox and Zucker (AKA, The Cox-Zucker Machine)...
