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An elliptic curve of rank ≥34\geq 34 and root number −1-1 over a quadratic field

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Dylan T. Costa ↗claimed, Alvaro Lozano-Robledo ↗

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Version 1 / Oct 16, 2026 / CC BY 4.0

Abstract

In this article, we find a quadratic twist over Q\mathbb{Q} of rank ≥2\geq 2 (and ≥3\geq 3 assuming parity) of an elliptic curve E/QE/\mathbb{Q} of rank ≥32\geq 32, recently found by Elkies, Klagsbrun, and Li. We use a family of quadratic twists of rank ≥2\geq 2 over Q(t)\mathbb{Q}(t) constructed by Mestre to search for twists EDE^D with root number −1-1, computing all root numbers without factoring the (very large) twisting parameter DD. As a result, we exhibit an explicit integer DD, with 411411 digits (a square times an integer with 381381 digits), such that EDE^D has root number −1-1 and two independent rational points. In particular, the rank of EE over Q(D)\mathbb{Q}(\sqrt{D}) is at least 3434, unconditionally, and at least 3535 if we assume the Parity Conjecture.

Provenance statement

We have used Claude Opus 5.5 for literature review, to help us write Magma code, and to help us search for the correct candidate squarefree D value such that E^D has the properties in main Theorem. Claude also helped putting together a first draft of this paper, but the authors rewrote it for readability. The ideas to find the twists came from the authors.

Tools used

Magma
Magma
Anthropic
Claude OpusVersion 5.5

References

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  1. Stewart, C. L.; Top, J. On ranks of twists of elliptic curves and power-free values of binary forms. J. Amer. Math. Soc., vol. 8, pp. 943–973. 1995

Version history

  1. v1Submitted by Alvaro Lozano-RobledoInitial depositCurrentOct 16, 2026