{
  "id": 6446,
  "curve_key": "32306059620820208496:-190260025755105724516693650456",
  "ainvs": [
    "0",
    "0",
    "1",
    "-673042908767087677",
    "220208363142483477449876910"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 134.83637840797095,
  "faltings_height": 9.157993548593195,
  "conductor": "773059754146050006342215950428641671043554055",
  "bad_primes": [
    "5",
    "11",
    "17",
    "826801876092032092344615989763253124110753"
  ],
  "discriminant": "-1436134847582702454438362945934893437844608467931296875",
  "regulator": "159599202082722.690741957858407373763348893609636151719512037892233065487",
  "points": [
    [
      "22129830067/49",
      "991789717898946/343"
    ],
    [
      "364105060743",
      "-219704509741803018"
    ],
    [
      "-1045042450637/1521",
      "1122822241923739778/59319"
    ],
    [
      "4194452872/9",
      "-75232589760721/27"
    ],
    [
      "18416593232/49",
      "-1546897936697994/343"
    ],
    [
      "1030699439543/3481",
      "-1406267255913829902/205379"
    ],
    [
      "7521128717687/6889",
      "16037752298439230274/571787"
    ],
    [
      "141297875245/81",
      "48105139531003766/729"
    ],
    [
      "451007924357/169",
      "290055106524618189/2197"
    ],
    [
      "517117750",
      "3232457710935"
    ],
    [
      "459197373",
      "2824218191577"
    ],
    [
      "31653201056139/58081",
      "54706960630139602142/13997521"
    ],
    [
      "646922963777403/900601",
      "8856987103582252705238/854670349"
    ],
    [
      "501346233",
      "2965338410622"
    ],
    [
      "-951028047",
      "-359994118863"
    ],
    [
      "67832259265/484",
      "120766657425755507/10648"
    ],
    [
      "-847198217",
      "13503221267222"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,39,48,56,59,83,90,104,122: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank 17, proved: the 17 listed points come from the point search and are independent, and the 2-descent upper bound (ellrank) is 17. Torsion trivial.",
  "created_at": "2026-11-03 14:18:38",
  "updated_at": "2026-11-03 14:18:38"
}