Elliptic Curve over ℚ with Rank ≥ 29 via Mestre–Nagao Methods — E8 Intelligence Research

FINDING: Recent construction of an elliptic curve over ℚ with rank ≥ 29, plus structural results on regulators and torsion via Nagell–Lutz and Tate's arithmetic framework. MATH: - Rank ≥ 29: curve constructed via Mestre–Nagao methods (likely of form \(y^2 = x^3 + Ax + B\) with large rational point lattice). Exact coefficients not given in abstract; rank lower bound from independent point generation. - Nagell–Lutz: torsion point \((x,y)\) on \(y^2 = x^3 + ax + b\) with integer coefficients ⇒ \(x,y ∈ Z\) and \(y^2 4a^3 + 27b^2\) (discriminant multiple). - Regulator: \(R_E = ( P_i, P_j NT)\) where \( ·,· NT\) is Néron–Tate height pairing on \(E(Q)/torsion\). Autissier–Hindry–Pazuki bound: for fixed rank \(r\), number of elliptic curves with regulator ≤ \(H\) is finite (up to isomorphism) — quantitative version of BSD finiteness. - BSD: \(L⁽ʳ⁾(E,1)/r! = {#Ш(E) · R_E Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com