# P2762: Rational points on y^2=x^6-x^2+1

- ID: `P2762`
- Reference: `rational-points-y2-x6-minus-x2-plus1`
- Page: https://theoremdb.org/statements/P2762
- Record maturity: Reviewed problem with recorded work

## Problem

Determine all affine rational pairs \((x,y)\in\mathbb Q^2\) satisfying \(y^2=x^6-x^2+1\).

### Problem setup

- **Definition.** 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.
- **Remark.** The polynomial x^6-x^2+1 is squarefree, so its smooth projective hyperelliptic model has genus two.

### What 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.

## Status

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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

The search checked the exact statement, parameters, equivalent terminology, and the sources listed in this packet. Database silence is treated only as bounded status evidence. A complete resolution must satisfy every acceptance condition in the canonical problem.

### Background and intake notes

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.

- Original intake status: UNKNOWN as of 2026-07-27. 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.

- Recorded example: The six points with x in {-1,0,1} and y in {-1,1} are affine rational solutions.

### Open directions

- **Route 1** (reported): 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. [1](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `rational-points-y2-x6-minus-x2-plus1`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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). The contributor selected the curve and wrote the rational-point classification after reviewing Mordell-Weil sieve methods. https://arxiv.org/abs/0906.1934
   - Also cited at Genus-two Mordell-Weil sieve and completeness methods
   - Also cited at Editorial research route recorded 2026-08-01
   - preprint; reference source; arXiv:0906.1934v2; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>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 https://arxiv.org/abs/1910.04653
   - preprint; reference source; arXiv:1910.04653v2; checked 2026-08-01
   - Source use: citation_only
   - 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.