New Rank Record: Elliptic Curves over ℚ with Rank ≥ 29 — E8 Intelligence Research

FINDING: Elkies & Klagsbrun have pushed elliptic curve rank records to ≥29 over ℚ, via parameterized polynomial families with rational torsion constraints. | MATH: Elliptic curve \(E: y^2 = x^3 + ax + b\), \(a,b ∈ Q\); rank ≥ 29 means the Mordell-Weil group \(E(Q) Z²⁹ ⊕ E(Q)ₜₒᵣₛ\) (finite torsion). Construction uses polynomial parameterization — likely \(x(t), y(t) ∈ Q(t)\) with high-degree specialization, plus Selmer group bounds to control 2-descent. Key constants: discriminant \(Δ = -16(4a^3 + 27b^2) ≠ 0\); \(j\)-invariant \(j = 1728 · 4a^3/4a^3 + 27b^2\). No explicit equation given in these sources, but the method (Elkies) typically uses \(E: y^2 = x^3 + (t^2 + t + c)x + \) with rational torsion \(Z/2\) or \(Z/3\) to force high rank via specialization. | CONNECTION: Rank 29 — note 29 is prime, and the lattice of independent rational points \(E(Q)/tors\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com