[#P2762] Rational points on y^2=x^6-x^2+1
Problem. Determine all affine rational pairs \((x,y)\in\mathbb Q^2\) satisfying \(y^2=x^6-x^2+1\).
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
Current 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]
1Records
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
2See also
- Unbounded numbers of integral points on global minimal elliptic curvesarithmetic geometry
- Birch and Swinnerton-Dyer rank conjecturearithmetic geometry
- Square-class collisions in the Pell-Lucas sequencediophantine equations
How to cite
TheoremDB contributors, “Rational points on y^2=x^6-x^2+1,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/rational-points-y2-x6-minus-x2-plus1This page as plain text: rational-points-y2-x6-minus-x2-plus1.md
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.