{
  "id": 6171,
  "curve_key": "70286476574315004991561:-18314157300546343749917010987983941",
  "ainvs": [
    "1",
    "0",
    "1",
    "-1464301595298229270658",
    "21196941319954761614647613982056"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 157.820599097468,
  "faltings_height": 11.040880280149485,
  "conductor": "809089619733908209365281749844742217326645985963890",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "29",
    "41",
    "991",
    "4219",
    "57446508608383991",
    "5555175449439498352409"
  ],
  "discriminant": "6840339107566432405906602665070256820936460738261572870895337500",
  "regulator": "1009490131502932927.1352641636715177401970230929572366706118946572550676259",
  "points": [
    [
      "-43156925049",
      "2002782602878939"
    ],
    [
      "-19539605874",
      "6507595354747429"
    ],
    [
      "-8559907320",
      "5753614070426527"
    ],
    [
      "17009028735",
      "1100639250353197"
    ],
    [
      "69750650535/4",
      "7859702956113161/8"
    ],
    [
      "18643372440",
      "614317751812792"
    ],
    [
      "18731765640",
      "583575272949592"
    ],
    [
      "25228198815",
      "558592624647667"
    ],
    [
      "25546540935",
      "679274506083997"
    ],
    [
      "26556134640",
      "1019245444635592"
    ],
    [
      "29577407970",
      "1939493978451892"
    ],
    [
      "29940783045",
      "2048163380009767"
    ],
    [
      "30415154982",
      "2190072277660924"
    ],
    [
      "39189111876",
      "4898793612919579"
    ],
    [
      "73932151650",
      "17805815517440692"
    ],
    [
      "-851610633299445/19321",
      "879403306443705533863/2685619"
    ],
    [
      "2148765942525/121",
      "1185693705487901717/1331"
    ],
    [
      "22058245425615/1369",
      "67698120687210724591/50653"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,9,26,30,45,62,91,102,127,136: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-30 14:39:40",
  "updated_at": "2026-10-30 14:39:40"
}