Record Rank 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. Now $\ge 30$, witnessed by an explicit curve $y^2 + xy = x^3 + a_4 x + a_6$ with $a_4$ of 63 digits and $a_6$ of 94, carrying thirty independent rational points.

Two tiers, and only the first is the record. Rank $\ge 30$ is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with $\Delta = 4.25$ gives an analytic rank of at most 31, and the root number is $+1$ so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial result because the open question is whether ranks are unbounded at all, which no single record answers. Superseded three days later by this project's own rank $\ge 31$ record (see the related entry); left unedited otherwise as a record of what was known at the time.