Numerical Exploration of the BSD Conjecture Through 12 Verified Elliptic Curves

This work presents a numerical investigation of the Birch and Swinnerton-Dyer (BSD) conjecture using SageMath. Twelve elliptic curves over the rational field are analyzed, spanning ranks from 0 to 3. Key invariants are computed and the BSD identity is numerically verified in each case. The paper also introduces symbolic implications relating to modular resonance, phantom points, and higher-dimensional harmonic structure.