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A computational study investigates BSD invariants in elliptic curves, revealing key constant behaviors and saturation rates.
Building on the p-adic conservation law for BSD invariants (DOI: 10.5281/zenodo.19534362), we conduct a systematic computational study of how Birch and Swinnerton-Dyer invariants behave under isogeny, using the full Cremona database (2,483,649 elliptic curves, conductor ≤ 500,000). Principal findings: (1) |Sha|·D is constant within every isogeny class (545,784 classes, zero exceptions); (2) Selmer saturation rate increases from 28% at rank 0 to 100% at rank 2; (3) |Sha|=1 for all 6,680 curves of rank ≥ 3; (4) Pr(v_p(|Sha|) varies) ~ 1/p⁴; (5) the 21 rank-2 exceptions are completely characterized. Includes 19 verification scripts.
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Tao Rui (2026) studied this question.
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