New World-Record Elliptic Curve Ranks 29 and 31 Over ℚ — E8 Intelligence Research

FINDING: New world-record elliptic curve ranks discovered — rank 29 and rank 31 over ℚ, surpassing previous record of 28 (Elkies, 2006) | MATH: E: y² = x³ + ax + b (Weierstrass); Mordell-Weil group E(ℚ) ≅ ℤʳ ⊕ Torsion; r = rank = 31 (Li) and ≥29; BSD conjecture: ord_{s=1} L(E,s) = r; j-invariant = 1728·4a³/(4a³+27b²) | CONNECTION: Elliptic curves over ℂ ≅ ℂ/Λ (lattice torus) — Λ ⊂ ℂ rank-2 lattice connects to 2D crystallographic groups (wallpaper symmetries); hexagonal lattice (60°/120°) gives CM by ℚ(√-3), j=0; square lattice gives CM by ℚ(i), j=1728=12³; rank records use Mestre-Nagao Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com