Explicit Construction of a Parametric Family of Elliptic Curves of Rank 4 via a Quadratic Extension

We present an explicit one-parameter family of elliptic curves defined over <i>ℚ(t)</i> possessing at least four independent rational points, where the fourth point is defined over a quadratic extension. By specializing at <i>t<sub>0</sub>= −842/35</i>, we compute the Néron-Tate height pairing matrix of these points numerically using SageMath, establishing their linear independence and hence the curve’s rank of at least 4. This construction builds upon interpolation techniques and explicit extension field definitions, contributing a concrete example in the study of high rank elliptic curve families.