Elliptic Curve Rank Leaderboard

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API

All numbers are exact integers or rationals (e.g. "49/4"), passed as strings to avoid precision loss. ainvs may be the short [a4, a6] or full [a1, a2, a3, a4, a6] Weierstrass form. The POST endpoints require an Authorization: Bearer <token> header — create a token on your profile page. The JSON downloads are public.

POST /api/submit

Submits a curve with a set of witness points. The points are checked to lie on the curve, and their independence is certified exactly by 2-descent quadratic characters at good primes (Cremona/Brumer; torsion handled via exact torsion generators), proving rank ≥ #points in E(ℚ) modulo torsion with no floating-point arithmetic in the decision. The response's independence.certificate lists the primes used and the F₂ matrix ranks; the Néron–Tate regulator is still reported as an informational diagnostic. On success the curve is recorded on the leaderboard (subject to the entry rule below); a new curve is attributed to you, while improving an already-recorded curve's rank bound updates its witness points but leaves the original submitter's credit in place (the improvement is credited to you in the curve's history and on your public page). Accepted curves and witness points are stored in the curve's global minimal model. Body: { ainvs, points }, where points is a list of [x, y] (an empty list submits the curve at rank ≥ 0).

The discriminant and Faltings height are recorded for every curve (neither needs the primes). Optionally include primes: the primes of bad reduction, equivalently the primes dividing the minimal discriminant. If they check out, the conductor is computed and recorded — no factoring needed — along with the verified prime list itself (deduplicated, sorted, extraneous good primes dropped). Re-submitting an existing curve with primes backfills the conductor even if the rank is unchanged.

Optionally include commentary: a string recorded as the curve's initial commentary, attributed to you. It is applied only when the curve has no commentary yet — if the curve already exists and already has commentary, this field is ignored (use POST /curve/:id/commentary to edit existing commentary). Commentary is plain text, with curve#123 links as described there.

curl -X POST https://elliptic-rank.icarm.cloud/api/submit \
  -H 'content-type: application/json' \
  -H 'authorization: Bearer erank_...' \
  -d '{
    "ainvs": ["0","0","1","-6349808647","193146346911036"],
    "points": [["49421","200114"], ["49493","333458"], ...],
    "primes": ["2","3","211",...],
    "commentary": "Found by Mestre (1982)."
  }'

Returns 200 with the result below, 422 if the submission is invalid (singular curve, point off curve, or not independent), 401 without a valid token, 413 if the request body is too large, 429 if your account is submitting too quickly, or 400 if the body isn't JSON. independence.rankLowerBound is the proven bound, canonical.key identifies the curve up to ℚ-isomorphism, and the leaderboard field reports the outcome — status is "created", "improved" (with previousRank), "unchanged" (a curve's record only changes when a witness proves a strictly higher rank), or "declined". A curve not yet on the board is added only if it places in the top 10 on some metric among curves of rank ≥ its own, either overall or among those with the same torsion subgroup (a "created" result carries "viaTorsion": true when only the latter applied); otherwise the submission is verified but not stored, and leaderboard is { "status": "declined", "rank", "limit", "torsion", "placement": { "naive", "faltings", "conductor", "disc" }, "torsionPlacement": { ... } } with each 1-based place overall and within the torsion subgroup (conductor is null when no primes were supplied, so include them if the conductor is the metric your curve is competitive on). Curves already on the board are never re-judged or removed.

{
  "ok": true,
  "curve":   { "ainvs": [...], "c4": "...", "c6": "...", "discriminant": "...", "nonsingular": true },
  "canonical": { "c4": "...", "c6": "...", "key": "304790815056:-166878443731135320" },
  "points":  [ { "point": ["49421","200114"], "onCurve": true }, ... ],
  "allPointsOnCurve": true,
  "independence": {
    "independent": true, "rankLowerBound": 12,
    "certificate": {            // the exact independence proof
      "primes": ["7","11","17", ...], "matrixRank": 12, "torsionRank": 0,
      "halvings": 0, "torsion": "[]"
    },
    "regulator": "...", "precisionDigits": 62,    // informational diagnostic
    "method": "..."
  },
  "height": { "naiveLogHeight": "79.3286..." },
  "faltingsHeight": "...",
  "conductor": "...",           // only if primes were supplied/recovered
  "badPrimes": ["2","3","211", ...],  // ditto: the verified primes, sorted
  "torsion": "[]",              // torsion structure (JSON invariant factors)
  "leaderboard": { "status": "created", "rank": 12 }
}

POST /api/curve/:id/primes

Backfill the conductor of an already-recorded curve from its primes of bad reduction — the programmatic equivalent of the primes form on a curve's page, and an alternative to re-submitting through /api/submit. (The discriminant and Faltings height are already recorded at submission.) Send { primes: [...] } with the primes of bad reduction (each a prime, together dividing the minimal discriminant to a unit), or { "mode": "auto" } to attempt bounded trial division (which gives up on hard factorizations). No factoring of large discriminants is performed otherwise.

curl -X POST https://elliptic-rank.icarm.cloud/api/curve/123/primes \
  -H 'content-type: application/json' \
  -H 'authorization: Bearer erank_...' \
  -d '{ "primes": ["2","3","211"] }'   # or: { "mode": "auto" }

Returns 200 with the computed invariants below. If the conductor is already recorded it is a no-op — { "ok": true, "alreadyRecorded": true } (fetch /curve/:id.json for the stored values). Returns 422 if the primes are invalid or incomplete (with errors and note), 404 if there is no such curve, 401 without a valid token, 413 if the body is too large, or 400 if the body isn't JSON or the id isn't an integer.

{
  "ok": true,
  "id": 123,
  "alreadyRecorded": false,
  "conductor": "...",
  "badPrimes": ["2","3","211"]
}

GET /api/curves/missing-primes

The curves whose primes of bad reduction (and so conductor) are not yet recorded, by id: { "curves": [{ "id": 12345, "discriminant": "76582385085647565163089..." }] }. Each discriminant is the minimal discriminant, so its prime divisors (ignoring the sign) are exactly the primes to send to /api/curve/:id/primes above. No auth required.

GET /database.json

The entire database as one JSON download. The top level carries attribution metadata (source, url, maintainer, acknowledgement, documentation) so a shared copy still says where it came from, then count and curves, each curve with its global-minimal a-invariants, transformed witness points, discriminant (the minimal discriminant), rank lower bound, naive height, Faltings height, and (when recorded) conductor, bad_primes (the verified primes of bad reduction — the factorization of the conductor's support), submitter, commentary, and history: later contributions by other users, oldest first, each { kind, user, at } with kind either "rank_improved" (also carrying old_rank and new_rank) or "primes_recorded". No auth required.

GET /curve/:id.json

A single curve as JSON — the same shape as one entry of the database.json curves array. No auth required.

POST /curve/:id/commentary

Edit a curve's commentary. Form-encoded content; an empty value clears it. Each edit is kept in the curve's commentary history.

Commentary is plain text (HTML is shown literally), except that curve#123 — no space — is displayed as a link to curve 123.

curl -X POST https://elliptic-rank.icarm.cloud/curve/123/commentary \
  -H 'authorization: Bearer erank_...' \
  --data-urlencode 'content=Found by Mestre (1982).'