{"id":"hexagon/2610.00280","pdf":"https://hexagonmath.org/pdf/2610.00280v1","url":"https://hexagonmath.org/abs/2610.00280v1","tags":["msc:11G05","msc:14H52","elliptic curve","parity conjecture","quadratic field","rank"],"score":0,"title":"An elliptic curve of rank $\\geq 34$ and root number $-1$ over a quadratic field","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Dylan T. Costa","Alvaro Lozano-Robledo"],"reviews":0,"subject":"math.NT","summary":"In this article, we find a quadratic twist over $\\mathbb{Q}$ of rank $\\geq 2$ (and $\\geq 3$ assuming parity) of an elliptic curve $E/\\mathbb{Q}$ of rank $\\geq 32$, recently found by Elkies, Klagsbrun, and Li. We use a family of quadratic twists of rank $\\geq 2$ over $\\mathbb{Q}(t)$ constructed by Mestre to search for twists $E^D$ with root number $-1$, computing all root numbers without factoring the (very large) twisting parameter $D$. As a result, we exhibit an explicit integer $D$, with $411$ digits (a square times an integer with $381$ digits), such that $E^D$ has root number $-1$ and…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Magma","Claude Opus"],"discussed":0,"published":"2026-10-16","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}