A Rank-$31$ Record for an Elliptic Curve over $\mathbb{Q}$

How large can the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$ be? Whether ranks are unbounded is open, and progress is measured by explicit records, tabulated by Dujella: rank $\ge 28$ from 2006, raised to $\ge 29$ by Elkies and Klagsbrun in 2024, and to $\ge 30$ three days before this one by the same team (see the related entry). Now $\ge 31$, witnessed by an explicit curve $y^2 + xy + y = x^3 + x^2 + a_4 x + a_6$ with $a_4$ of 67 digits and $a_6$ of 99, carrying thirty-one independent rational points.

Two tiers, and only the first is the record. Rank $\ge 31$ is unconditional: 31 explicit points, independence asserted via the leaderboard's stated general practice of exact 2-descent (not reproduced here - see the verification note). Rank exactly 31 is conditional on GRH and BSD, per the submitters' commentary, in the same style as the sibling record's Bober-bound argument; no numeric derivation has been published for this curve specifically. The entry is a partial result because the open question - whether ranks are unbounded at all - remains unanswered by any single record.