{
  "id": 6106,
  "curve_key": "33228790796736441921:-162762686693643251124420209505",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-692266474932009207",
    "188382739401820381608488655"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 134.84984510008118,
  "faltings_height": 9.231803131164824,
  "conductor": "152829269309409686331640851550666966969954",
  "bad_primes": [
    "2",
    "11",
    "13",
    "31",
    "17237679822852434731743836177607372769"
  ],
  "discriminant": "5901598094299718325614404086495305511448419874263334912",
  "regulator": "15633324452610939415.96172021830679064297761258593285682019230566327098",
  "points": [
    [
      "-512217203",
      "20213739748522"
    ],
    [
      "307325235",
      "2158151671710"
    ],
    [
      "-677284929845/729",
      "107130404752103962/19683"
    ],
    [
      "9688729299/49",
      "2639775000742706/343"
    ],
    [
      "126974487019/5625",
      "5545165283374184122/421875"
    ],
    [
      "2181073429743/3481",
      "160616797939134378/205379"
    ],
    [
      "5662946311091/5329",
      "9938849992388824430/389017"
    ],
    [
      "90992767659119/42025",
      "810005677099750468262/8615125"
    ],
    [
      "125904878114627/36481",
      "1374396023455195081570/6967871"
    ],
    [
      "252816002820235/38809",
      "3988255825057070374950/7645373"
    ],
    [
      "1651083933394275/7187761",
      "124115356097295513700030/19270387241"
    ],
    [
      "11166548008615611/5929225",
      "1076965026468039360492586/14437662875"
    ],
    [
      "56708530601647935/4592449",
      "13474241984173546531937138/9841618207"
    ],
    [
      "175554716294162955/130256569",
      "61357376043046636795397030/1486618221997"
    ],
    [
      "240902438741723403/5368489",
      "118218802604233753831034870/12438789013"
    ],
    [
      "396698128161729619/1238769",
      "249855192001693803354647182/1378749897"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,13,29,52,54,55,69,71,106,125: 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": "2026-10-29 02:29:32",
  "updated_at": "2026-10-29 02:29:32"
}