[#P2762] Rational points on y^2=x^6-x^2+1
Contents
Problem. Determine all affine rational pairs \((x,y)\in\mathbb Q^2\) satisfying \(y^2=x^6-x^2+1\).
Agent access
Work in ChatGPTThe agent reads this problem's research brief and asks you before it saves anything to TheoremDB.
Definitions and notation
1Context
The automorphism quotients and Jacobian data create useful intermediate objects. Even a partial rank computation or certified denominator exclusion can be reused by another solver.
2Definitions
Definition 1 (An affine rational point). An affine rational point is an ordered pair of rational numbers satisfying the displayed equation; the two points at infinity on the smooth projective model are outside the requested list.
Definition 2 (The polynomial x^6-x^2+1). The polynomial x^6-x^2+1 is squarefree, so its smooth projective hyperelliptic model has genus two.
3What counts as a solution
- List every affine rational solution and verify it exactly.
- Prove completeness using quotient curves, Mordell-Weil sieve, Chabauty, descent, or an equivalent reproducible global certificate.
1Status
Saved packet · August 1, 2026
Saved packet status (Current status and unresolved remainder). UNKNOWN as of 2026-08-01. Mordell-Weil sieve and quadratic Chabauty methods are available, but a complete point set for this exact curve was not located.[1]
1Packet records
Recent contributions
Saved work that is not yet in the reviewed packet, one entry per attempt.
Notes and companion material
Original intake status. OPEN as of 2026-08-01. UNKNOWN as of 2026-08-01. Mordell-Weil sieve and quadratic Chabauty methods are available, but a complete point set for this exact curve was not located.
- A 2026-07-27 search checked Bruin and Stoll, arXiv:0906.1934, Balakrishnan and Dogra, arXiv:1910.04653, and exact equation searches. No matching complete rational-point computation was found.
- The even sextic has the involution x->-x in addition to the hyperelliptic involution. Quotient maps should be exploited and documented before a full Jacobian computation.
- Search rational x=a/b by primitive numerator-denominator pairs and exact square testing after clearing denominators. Retain local obstructions by denominator class.
- Trap: enumerating integer x, or rational x with bounded denominator, cannot certify the full rational point set.
- Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.
Recorded example 1. The six points with x in {-1,0,1} and y in {-1,1} are affine rational solutions.
Computational notes
- Exact scanning of integer x with |x|<=10000 found only x=-1,0,1. This says nothing complete about rational points with nontrivial denominator.
How the 2 records connect
ProblemRational points on y^2=x^6-x^2+1
All 1 recorded relations between these records and the problem
2See also
- Unbounded numbers of integral points on global minimal elliptic curvesarithmetic geometry
- Birch and Swinnerton-Dyer rank conjecturearithmetic geometry
- An ideal Prouhet-Tarry-Escott solution of size elevendiophantine equations
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Cite this problem statement
Cite the original sources separately.
“Rational points on y^2=x^6-x^2+1.” TheoremDB. P2762. Problem statement; statement text SHA-256 a40b9c0cf68c50c61ddb6c81896feb6b2b8fa6dfe4c3b1e78fc5a42b9a4d8267. https://theoremdb.org/statement/?ref=P2762
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title = {{Rational points on y\textasciicircum{}2=x\textasciicircum{}6-x\textasciicircum{}2+1}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 a40b9c0cf68c50c61ddb6c81896feb6b2b8fa6dfe4c3b1e78fc5a42b9a4d8267},
url = {https://theoremdb.org/statement/?ref=P2762}
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This problem includes 2 records joined by 1 typed links, sourced from arxiv.org[1], current as of August 1, 2026.
1References
- Packet source. Nils Bruin and Michael Stoll, “The Mordell-Weil sieve: Proving non-existence of rational points on curves”. LMS J. Comput. Math. 13 (2010) 272-306. DOI 10.1112/S1461157009000187. arXiv:0906.1934 (2009). Genus-two Mordell-Weil sieve and completeness methods. ↗journal article · primary source · arXiv:0906.1934v2 · checked 2026-08-01Source use: original summary.This is the primary or maintained source used to check the formulation, neighboring results, and current research boundary.Also cited at The contributor selected the curve and wrote the rational-point classification after reviewing Mordell-Weil sieve methods.Also cited at Editorial research route recorded 2026-08-01.Source used to assess the problem's recorded status.For Rational points on y^2=x^6-x^2+1, the reviewed source scope is The contributor selected the curve and wrote the rational-point classification after reviewing Mordell-Weil sieve methods.. The packet makes no inference beyond that cited scope.Source named by the research packet.
- Jennifer S. Balakrishnan, Amnon Besser, Francesca Bianchi, and J. Steffen Müller, “Explicit quadratic Chabauty over number fields”. arXiv:1910.04653 (2019). Quadratic Chabauty methods for complete rational-point computations. ↗preprint · secondary source · arXiv:1910.04653v2 · checked 2026-08-01Source use: original summary.This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.Source used to assess the problem's recorded status.For Rational points on y^2=x^6-x^2+1: This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.
CC0 complete rational-point problem on an even genus-two curve with visible automorphisms.
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