{
  "id": 6022,
  "curve_key": "50630269792622356641:-359213680907724481940686936689",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-1054797287346299097",
    "415756575388380435193851321"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 136.1132438657134,
  "faltings_height": 9.156678064592585,
  "conductor": "1653535489753497367005066448469162613580430",
  "bad_primes": [
    "2",
    "5",
    "37",
    "139",
    "42923",
    "294563",
    "442953261827",
    "5740779015797887"
  ],
  "discriminant": "435411610818674532275561305317217450068180556288000000",
  "regulator": "74858446729909.35099432563094123847526469104775526969200245848746945598",
  "points": [
    [
      "-954076489",
      "23530337745244"
    ],
    [
      "-375345609",
      "27546327007724"
    ],
    [
      "-75772809",
      "22254171834524"
    ],
    [
      "267753511",
      "12350034851244"
    ],
    [
      "403866151",
      "7458578232364"
    ],
    [
      "561961971",
      "684488537004"
    ],
    [
      "619766511",
      "295295231244"
    ],
    [
      "2497720539/4",
      "6104734445797/8"
    ],
    [
      "631806871",
      "1237835569404"
    ],
    [
      "636175461",
      "1480512459594"
    ],
    [
      "725406511",
      "5684640066244"
    ],
    [
      "817839911",
      "10005758777644"
    ],
    [
      "1150596391",
      "26931803559724"
    ],
    [
      "1234911011",
      "31565527531244"
    ],
    [
      "51607100191/81",
      "1116316338482476/729"
    ],
    [
      "151042193959/289",
      "13216789721474252/4913"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,10,24,32,36,47,49,51,72: 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 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-28 11:28:37",
  "updated_at": "2026-10-28 11:28:37"
}