{
  "id": 186,
  "curve_key": "20802047344590934454546241:-93310437331621014194286256464566516961",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-433375986345644467803047",
    "107998191356151184499232541284273119"
  ],
  "rank_lower_bound": 16,
  "naive_height": 174.8912809312466,
  "faltings_height": 12.461662233392545,
  "conductor": "80641417935190941169492920447777598088517349263967556694390",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "13219",
    "32259473509",
    "8349520517389",
    "14802985393449629740213309277"
  ],
  "discriminant": "170562405115020174321241730237321859944276566942438388072716846080000000",
  "regulator": "139758098201894122.2826035804257146048295377794883418723209203275293760",
  "points": [
    [
      "117823316540065899/105625",
      "34543457264653001082310582/34328125"
    ],
    [
      "3714470393587",
      "7053214591294192926"
    ],
    [
      "180092834680363333/21904",
      "76188679591183554465639687/3241792"
    ],
    [
      "10159084638447",
      "32313984927375252466"
    ],
    [
      "413500337987667",
      "8408397657985911069886"
    ],
    [
      "8822791558227",
      "-26135518036715157314"
    ],
    [
      "1290399889827",
      "-1302865125960336914"
    ],
    [
      "796410792327",
      "-517680291105402914"
    ],
    [
      "156319180011123/289",
      "-876615168560382378082/4913"
    ],
    [
      "1577789313211767/3721",
      "-4942957593625439902634/226981"
    ],
    [
      "4460191705964563/10609",
      "-11374202741656054438878/1092727"
    ],
    [
      "16971071132626147/40401",
      "70123120568310244525166/8120601"
    ],
    [
      "477541992066122787/1018081",
      "91420772927296554619724494/1027243729"
    ],
    [
      "481359300087",
      "104512157193136426"
    ],
    [
      "144856432632003/289",
      "634955008440682893518/4913"
    ],
    [
      "441065116592403/529",
      "6949883465755085952962/12167"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2901/10: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.",
  "created_at": "2026-06-25 16:02:31",
  "updated_at": "2026-07-01 22:39:38"
}