{
  "id": 6073,
  "curve_key": "73589609931292365001:-644865570133943593596428491525",
  "ainvs": [
    "1",
    "0",
    "1",
    "-1533116873568590938",
    "746372187527267567939224472"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 137.27767877503177,
  "faltings_height": 9.337986524567963,
  "conductor": "1027456936062822754905350724274248877209558",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "17",
    "61",
    "4726157",
    "64824132232123650171832733443"
  ],
  "discriminant": "-10030191822290724909508782535593498730100636153928498108",
  "regulator": "16456381910784143844.79213298261212750089522371632206090135644840239513",
  "points": [
    [
      "81041959408",
      "23068202032447895"
    ],
    [
      "352350934171",
      "209151073135865873"
    ],
    [
      "-631842771513286657994/1268342206849",
      "53188240969226807520626084686724/1428415871748791743"
    ],
    [
      "-140677916659964/249001",
      "4702266420558521514617/124251499"
    ],
    [
      "-135550367748479/1324801",
      "45800464598213821711705/1524845951"
    ],
    [
      "6662264431/676",
      "475288112932935505/17576"
    ],
    [
      "104126726929/64",
      "25898598538946971/512"
    ],
    [
      "295029437299/625",
      "176672398557916793/15625"
    ],
    [
      "1313700671383/3721",
      "3582453922356021635/226981"
    ],
    [
      "1558586750491/1521",
      "940417785061664645/59319"
    ],
    [
      "1955265222793/9",
      "2734015736839582397/27"
    ],
    [
      "42659363395819/52441",
      "73563759618556830797/12008989"
    ],
    [
      "1126656107994433/2399401",
      "42378304595205757824479/3716672149"
    ],
    [
      "8302685671910320/23068809",
      "1720798425210022581267752/110799489627"
    ],
    [
      "32742610595314855/29041321",
      "3323573181057724871394467/156503678869"
    ],
    [
      "1691788583096874271/534811876",
      "2052895325547573766525493023/12368059444376"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,26,38,43,44,51,63,83,96: 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 21:14:49",
  "updated_at": "2026-10-28 21:14:49"
}