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April 14, 2026Open Access

A Systematic Computational Study of BSD Invariant Behavior under Isogeny: Selmer Saturation, Sha Vanishing, and the 1/p⁴ Law

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Authors

TRTao Rui

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Overview

A computational study investigates BSD invariants in elliptic curves, revealing key constant behaviors and saturation rates.

Key Points

  • This research aims to analyze the behavior of Birch and Swinnerton-Dyer (BSD) invariants under isogeny using a large database of elliptic curves.
  • Utilized the full Cremona database comprising 2,483,649 elliptic curves with conductors ≤ 500,000.
  • Conducted a systematic computational study focusing on BSD invariant behaviors.
  • Characterized exceptions in rank-2 scenarios.
  • Found that |Sha|·D remains constant for every isogeny class with no exceptions across 545,784 classes.
  • Observed an increase in Selmer saturation rates from 28% at rank 0 to 100% at rank 2.
  • Determined |Sha| equals 1 for all 6,680 curves of rank ≥ 3.
  • Introduced a relationship indicating Pr(v_p(|Sha|) varies) ~ 1/p⁴.
  • Completely characterized 21 rank-2 exceptions.

Cite This Study

Tao Rui (2026) studied this question.

synapsesocial.com/papers/69ddd9e1e195c95cdefd7413https://doi.org/10.5281/zenodo.19539164
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