Record-Breaking Elliptic Curve with Rank ≥ 29 Discovered via Explicit Construction — E8 Intelligence Research

FINDING: A new elliptic curve over ℚ with rank ≥ 29 has been discovered, breaking the previous record; explicit construction methods (2-isogenies, Heron-form families) are the active tools. MATH: - Elliptic curve: \(E: y^2 = x^3 + ax + b\), \(a,b \in \mathbb{Q}\). - Rank \(r\) = number of independent infinite-order rational points; regulator \(\text{Reg}(E) = \det(\langle P_i, P_j \rangle)\) where \(\langle \cdot,\cdot \rangle\) is the Néron–Tate height pairing. - For rank 29: the height pairing matrix is 29×29, positive definite, with entries \(h(P_i+P_j)-h(P_i)-h(P_j)\). - Elkies–Klagsbrun: rank records for curves with rational torsion — typically use \(E(\mathbb{Q})_{\text{tors}} \cong \mathbb{Z}/m\mathbb{Z}\) with \(m \in \{2,3,4,5,6,7,8,9,10,12\}\) (Mazur's theorem). - Heron-family construction: given \((A^2,B^2,C^2)\) with Heron area, the curve \(y^2 = x(x-A^2)(x-B^2)\) has rank ≥ 5 (arXiv:1501.03809). CONNECTION: - Height pairing matrix is a Gram matrix of a latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com