{
  "id": 6026,
  "curve_key": "29746956448558455696:-161466481983577640774630558400",
  "ainvs": [
    "0",
    "0",
    "0",
    "-619728259344967827",
    "186882502295807454600266850"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 134.51777548920268,
  "faltings_height": 9.04382070828085,
  "conductor": "3055278382854708194585081652179409474600432",
  "bad_primes": [
    "2",
    "3",
    "7",
    "17",
    "10930743779",
    "1812379369747177366245439267"
  ],
  "discriminant": "145314929292065166595657696694406762076570337306330112",
  "regulator": "39087593875083.13516755449762418728637523422598578344441381345565295848",
  "points": [
    [
      "-52729959",
      "14812634305308"
    ],
    [
      "325288773",
      "4439771220864"
    ],
    [
      "349023099",
      "3619394242674"
    ],
    [
      "388357821",
      "2186094777888"
    ],
    [
      "409814949",
      "1317694271376"
    ],
    [
      "480129193",
      "119144852036"
    ],
    [
      "494626011",
      "1166815392078"
    ],
    [
      "494692413",
      "1170064198464"
    ],
    [
      "495585627",
      "1213444922802"
    ],
    [
      "532265949",
      "2795879640624"
    ],
    [
      "571646857",
      "4406770289852"
    ],
    [
      "690919129",
      "9408715884316"
    ],
    [
      "831807037",
      "15713621963648"
    ],
    [
      "7032796125",
      "586235543918640"
    ],
    [
      "2435578641/4",
      "47519338790847/8"
    ],
    [
      "2709462913/9",
      "141839468185996/27"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,18,20,42,50,58,59,68,76: 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 11:29:22",
  "updated_at": "2026-10-28 11:29:22"
}