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Arithmetic geometry · elliptic curves · Mordell–Weil rank

Elliptic Curves over ℚ of Rank at Least 30

Collaboration beta

Can one exhibit a nonsingular elliptic curve over ℚ with Mordell–Weil rank at least 30, together with an exact certificate? The maintained record is rank at least 29, so the target remains open.

rankE(ℚ)≥30
Known results and sources
A smooth elliptic curve over Q sits beside a rank-at-least-30 question, with the known 29 benchmark separated from an unresolved thirtieth direction.
The maintained record is rank at least 29; constructing and exactly certifying rank at least 30 remains open.

Research problem

Exact mathematical statement

Construct a nonsingular elliptic curve E/ℚE/\mathbb Q for which

rankE(ℚ)≥30.\operatorname{rank} E(\mathbb Q)\ge 30.

An acceptable explicit certificate consists of a nonsingular Weierstrass equation, thirty exact rational points on the curve, exact point-substitution checks, and a rigorous proof that those points generate a subgroup of rank 30.

Problem infographic

Problem at a glance

A non-coordinate three-panel landscape shows a smooth elliptic-curve silhouette without point markers, an abstract rank-29 ribbon bundle beside an unresolved thirtieth direction, and the exact rank-at-least-30 certificate requirements.
The problem requires an explicit curve over Q, thirty rational points, exact point checks, and a rigorous rank-30 independence certificate; none is yet recorded.

Current mathematical picture

Where work on Elliptic Curves over ℚ of Rank at Least 30 stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureBlind enlargement of previously bounded direct point searches

The no-repeat ledger rejects treating an approximate rank-29 height matrix, one prime, or a function-field construction without exact rational specialization as a rank-30 certificate. The K3 reconstruction route remains a source-proposed program if a branch-safe finite-field solution is isolated, independently cross-checked, lifted, and verified over ℚ(T).

Route status · Narrowed route
Main reductionCurrent reduction

The source prioritizes reconstructing a rank-17 K3 fibration from the audited rank-29 specialization, then seeking a specialization with one additional independent direction.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeSolve the materialized branch-gauged 38-variable compatibility core over 𝔽₇₃ without discarding stabilizer or denominator branches.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Elliptic Curves over ℚ of Rank at Least 30 in numbers

3.2kretained lines of mathematical investigation3,197 in the current working snapshot
Argument development
2,586 · 81%
Explored or eliminated routes
87 · 3%
Computational analysis
298 · 9%
Open obligations
94 · 3%
Definitions and setup
132 · 4%
7selected mapped statements1routes investigated5open questions5contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 15, 2026
Source recordedAug 15, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Elliptic Curves over ℚ of Rank at Least 30A selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can one construct an elliptic curve over ℚ with thirty independent rational points? — Depends on missing premiseCan one construct anelliptic curve over ℚ withthirty…Current reduction — Depends on missing premiseCurrent reductionExact rank-30 certificate — Depends on missing premiseExact rank-30 certificateClosing target — Depends on missing premiseClosing targetDirect bounded searches exhausted — Depends on missing premiseDirect bounded searchesexhaustedExact rank-29 starting architecture — Depends on missing premiseExact rank-29 startingarchitectureNo rank-30 curve produced — Depends on missing premiseNo rank-30 curve producedBlind enlargement of previously bounded direct point searches — stoppedBlind enlargement ofpreviously bounded directpoint…Solve the materialized branch-gauged 38-variable compatibility core over 𝔽₇₃ without discarding stabilizer or denominator branches. — OpenSolve the materializedbranch-gauged 38-variablecompatibility…Validate every non-isotrivial finite-field solution and cross-check its corresponding branch independently over 𝔽₈₃. — OpenValidate everynon-isotrivial finite-fieldsolution…Lift an isolated nonsingular finite-field point and recognize exact rational coefficients for the surface and its sections. — OpenLift an isolated nonsingularfinite-field point andrecognize…Reduced compatibility core unsolved — OpenReduced compatibility coreunsolvedExact independence still required — OpenExact independence stillrequired
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeBlind enlargement of previously bounded direct point searches

The no-repeat ledger rejects treating an approximate rank-29 height matrix, one prime, or a function-field construction without exact rational specialization as a rank-30 certificate. The K3 reconstruction route remains a source-proposed program if a branch-safe finite-field solution is isolated, independently cross-checked, lifted, and verified over ℚ(T).

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Solve the materialized branch-gauged 38-variable compatibility core over 𝔽₇₃ without discarding stabilizer or denominator branches.Suggested move: Compute the actual ideal or component structure sparsely and retain exact branch and duplicate-removal certificates.
Ready to work on
02
Validate every non-isotrivial finite-field solution and cross-check its corresponding branch independently over 𝔽₈₃.Suggested move: Verify all sections, chord relations, fiber data, minimality, and the determinant-948 function-field height matrix.
Ready to work on
03
Reduced compatibility core unsolved

The current work labels the 38-variable triangular core generated-exact but says the branch-gauged ideal remains unsolved.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Exact independence still required

Any future candidate still needs a rank-30 Kummer matrix or an exact certified height matrix with required saturation.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Lift an isolated nonsingular finite-field point and recognize exact rational coefficients for the surface and its sections.Suggested move: Perform multivariate 73-adic lifting, rational reconstruction, and exact substitution and height verification over ℚ(T).
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen problem

A maintained university-hosted record table lists rank at least 29 as the current record, with the corresponding curve and 29 independent points. It does not list a rank-at-least-30 example, so the exact construction target remains open.

[1][2]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Computational resultThe maintained record table reports an Elkies–Klagsbrun curve with rank at least 29 and lists 29 independent points.[1][2]
  2. Authoritative summaryElkies produced a curve over Q with 28 independent rational points.[1][3]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusElliptic Curves over ℚ of Rank at Least 30
Solved special caseExplicit curve over Q of rank at least 29

A concrete curve and 29 independent points establish the current rank-at-least-29 benchmark, one rank below the present target.

[2]
Related problemExact rank of the Elkies rank-28 curve under GRH

A conditional exact-rank result for a rank-28 curve is evidence about a neighboring high-rank example, not a rank-at-least-30 construction.

[4]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • dataset · source linked; not reproduced by ProofAtlasRank-29 curve and independent points

    The maintained page lists the curve and 29 independent points; it is a benchmark dataset, not a rank-30 certificate.

    [2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal definition of Mordell–Weil rank and exact rational points for the chosen curve model.
  • Formalization targetA checked nonsingularity and point-membership certificate for a future rank-30 candidate.
  • Formalization targetA checked independence or saturation certificate proving rank at least 30.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements5 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction1 of 71
  • lemma2 of 72
  • equivalence1 of 71
  • negative result2 of 72
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementCan one construct an elliptic curve over ℚ with thirty independent rational points?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact rank-30 certificateintermediate
  • retained route statementNo rank-30 curve producedintermediate
  • retained route statementExact rank-29 starting architectureintermediate
  • retained route statementDirect bounded searches exhaustedintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureBlind enlargement of previously bounded direct point searchesreported failure
  • Research targetSolve the materialized branch-gauged 38-variable compatibility core over 𝔽₇₃ without discarding stabilizer or denominator branches.open
  • Research targetValidate every non-isotrivial finite-field solution and cross-check its corresponding branch independently over 𝔽₈₃.open
  • Research targetLift an isolated nonsingular finite-field point and recognize exact rational coefficients for the surface and its sections.open
  • Research targetReduced compatibility core unsolvedopen
  • Research targetExact independence still requiredopen
  • Narrowed routeBlind enlargement of previously bounded direct point searchesThe no-repeat ledger rejects treating an approximate rank-29 height matrix, one prime, or a function-field construction without exact rational specialization as a rank-30 certificate. The K3 reconstruction route remains a source-proposed program if a branch-safe finite-field solution is isolated, independently cross-checked, lifted, and verified over ℚ(T).
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeSolve the materialized branch-gauged 38-variable compatibility core over 𝔽₇₃ without discarding stabilizer or denominator branches.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointSolve the materialized branch-gauged 38-variable compatibility core over 𝔽₇₃ without discarding stabilizer or denominator branches.

Elliptic Curves over ℚ of Rank at Least 30 · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Can one exhibit a nonsingular elliptic curve over ℚ with Mordell–Weil rank at least 30, together with an exact certificate? The maintained record is rank at least 29, so the target remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
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Sources and references4 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    History of elliptic curves rank recordsauthoritative webpage · Andrej Dujella · University of Zagreb · accessed Aug 15, 2026
  2. 2
    Rank ≥ 29software or dataset · Noam D. Elkies, Zev Klagsbrun, Andrej Dujella · University of Zagreb · 2024 · accessed Aug 15, 2026
  3. 3
    Three lectures on elliptic surfaces and curves of high rankpreprint · Noam D. Elkies · arXiv · 2007 · ARXIV 0709.2908 · accessed Aug 15, 2026
  4. 4
    The Elkies Curve has Rank 28 Subject only to GRHpreprint · Zev Klagsbrun, Travis Sherman, James Weigandt · arXiv · 2016 · ARXIV 1606.07178 · accessed Aug 15, 2026

Important qualifications

  • The bounded pass checked bibliographic identity, current status, and representative official special-case sources; it was not an exhaustive bibliography or priority review.
  • The private packet and its URLs were not used as external authority. No submitted URL or attachment was fetched, executed, compiled, or rendered.
  • No formal proof of the unrestricted statement was identified in this bounded pass; absence from the pass is not proof of absence.

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