Elliptic Curve over ℚ Achieves Record-Breaking Rank 31 — E8 Intelligence Research

FINDING: Samuel Li discovered an elliptic curve over ℚ with rank 31, breaking the previous record (rank 28 by Elkies 2006, rank 29 announced 2024) — the highest known rank for an elliptic curve over the rationals, demonstrating unbounded growth of Mordell-Weil groups. MATH: E: y² + a₁xy + a₃y = x³ + a₂x² + a₄x + a₆ over ℚ | rank(E/ℚ) = 31 | Mordell-Weil: E(ℚ) ≅ ℤ³¹ ⊕ Torsion | Height pairing matrix 31×31 with regulator Reg(E) ≈ 10⁴⁰ scale | j-invariant and coefficients are massive integers (100s of digits) | BSD conjecture predicts L⁽³¹⁾(E,1)/31! = Reg(E)·|Ш|·∏cₚ/|E(ℚ)ₜₒᵣₛ|² CONNECTION: Elliptic curves = ℂ/Λ (complex torus) where Λ = ℤω₁ ⊕ ℤω₂ is a rank-2 lattice — crystallographic symmetry of 2D latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com