Stage-2 universe
Canonical-coarse recorded graph
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Mathematical companion to a preprint in preparation
Modular covers of positive-rank elliptic curves
A dated inventory of LMFDB-recorded parent-cover endpoint pairs and route witnesses, with verified examples clearly separated from database observations.
Strict-view extract
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Completeness is relative to a stated database universe.
At a glance
The strict research view
Counts refer to distinct source–target endpoint pairs after the published filters. Quartic route witnesses are counted separately from endpoints.
Stages 1–2
The graph audit, then the strict view
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Published strict view
Linked recorded-rank-[1,1], high-gonality rows
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Audit boundary
- Graph-audit extract
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- Audit result
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- Fine-refinement closure
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- Audit window
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- Schema
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The downloadable metadata carries the complete statement and the evidence IDs.
Included in the audit
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Outside the claim
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Evidence IDs load with the snapshot.
Read the full scope and search-method note
Checked in depth
Serious candidates
Each card states the strongest formal status justified today. “Featured in the paper” is a separate editorial marker and never upgrades a proof status.
Search the collection
Strict filtered cover inventory
Every row is an endpoint pair remaining after the linked-target, recorded rank-bounds [1,1], genus, and gonality filters. Open its details for the exact index witness, provenance hashes, and verification state.
Swipe or scroll horizontally to see every column.
| Recorded degree | Source C | Genus | gonQ̅(C) | Construction | Modular target | Elliptic target | Proof status | Open details |
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No endpoint pairs match these filters.
Quartic factorizations
Recorded two-step witnesses
At the graph level, a chain C → M → E of two recorded index-two parent relations is retained as a degree-four route witness. It is not by itself proof of an explicit geometric map. Different witnesses can share an endpoint pair; parent positions and conjugators keep their identities exact.
Swipe or scroll horizontally to inspect the full witness.
| Witness | Source C | Intermediate M | Modular target | Parent positions | Conjugators | Evidence | Open details |
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Stage 3
Unlinked genus-one target ledger
Target labels are grouped across their incident degree-3 and degree-5 boundary records. Resolved rows record a pointed elliptic model and algebraic rank; they do not certify any incident parent map.
Swipe or scroll horizontally to inspect target evidence.
| Modular target | Incident degrees | Sources | Origin state | Model | Rank | Elliptic target | Equation / point | Open details |
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Stages 4–5
From a database row to a theorem
The proof ledger uses explicit enum states: verified, not verified, not checked, and not applicable. Only selected serious candidates have a completed ledger.
Map degree
Exact geometric degree, beyond a bare index ratio.
Target model
A pointed Weierstrass model for the actual target.
Algebraic rank
Certified positive rank; analytic rank is not substituted.
Geometric gonality
An independently justified lower bound over Q̅.
Genus hypothesis
The strict inequality required by the theorem.
Discriminant divisor
The trace-discriminant divisor on the smooth cover.
Derangement
The fixed-point-free condition on the odd support.
Rational-fibre reduction
The exact route controlling lower-degree fibres.
Density theorem
Applicability of the stated field-density conclusion.
LMFDB-listed — database observation only
Model-verified — exact target/model work recorded
Branch-verified — Stage 5 branch ledger recorded
Theorem-ready — all project gates green and signed
Paper-featured — editorial tag, independent of status
How the inventory is made
Scope and evidence
This is an exact report on specified fields in a dated database snapshot. It is a discovery resource; proof claims live in separate, evidence-linked candidate records.
Target policy
Strict linked rows retain the recorded elliptic identification and recorded rank bounds [1,1]. Missing target data remains missing.
Gonality filter
The lower recorded geometric-gonality bound is at least 4, 5, or 6 for degrees 3, 4, or 5 respectively.
Graph witnesses
Prime-degree rows use recorded parents. Quartic composites retain every recorded two-by-two chain separately.
Immutable provenance
Normalized and raw-record hashes, source files, queries, and evidence IDs travel with the public edition.
Read, audit, and reuse
Data and documentation
Start with the mathematical guide, then choose the normalized tables or provenance package needed for an independent audit.
For mathematicians
A friendly route into the repository
Machine-readable
Public editions
JSON indexEndpoint search indexCompact endpoint rows, audit summaries, and on-demand evidence references
JSONProof recordsStage 4 ledgers and Stage 5 candidate dossiers
JSON indexQuartic witness search indexEvery retained route identity, with full evidence loaded from bounded shards
CSV editions and supplementary records
Supplementary records
Unlinked boundary and workbook
Boundary rows are target-only records, not strict cover examples. Some targets now have independently resolved pointed elliptic models and rank; no incident parent map is promoted by that resolution. The degree ledgers remain separate so unresolved models and untouched cover gates stay visible.
Before citation
Check the validation report
The report reconciles counts, schema, file hashes, proof-record references, and the route/endpoints split.
Citation
Diana Mocanu and George Țurcaș, New Algebraic Points on Covers of Elliptic Curves, preprint in preparation.