{
  "id": 207,
  "curve_key": "160382049829531714473522201612289:-2038251840626530708856061601274460704106400167937",
  "ainvs": [
    "1",
    "0",
    "0",
    "-3341292704781910718198379200256",
    "2359087778502928782734938491871399298406363136"
  ],
  "rank_lower_bound": 12,
  "naive_height": 222.4723539266345,
  "faltings_height": 16.360697581894815,
  "conductor": "877030824470138245488472257225334285674448305325975329534240593693229985430",
  "bad_primes": [
    "2",
    "5",
    "7",
    "13",
    "17",
    "19",
    "157",
    "1153",
    "931690673197751",
    "1040689837985892942452948677587572182667636492893"
  ],
  "discriminant": "-16816563281216520792334587550778720192475526428103707017379511232474985349250763174857420800",
  "regulator": "296423040996438.87421831030116833563744437043438121358288765559",
  "points": [
    [
      "10124204966610246124/9",
      "-32213715633639840416999764496/27"
    ],
    [
      "8018456777604735778/81",
      "-22701924064273317034280911460/729"
    ],
    [
      "258291942189627096/169",
      "-63006127372568334146060224/2197"
    ],
    [
      "87243969374310784555424/78552769",
      "-2963040146283390716652256622504576/696213191647"
    ],
    [
      "157375705483764349/144",
      "-6177158242078167048846049/1728"
    ],
    [
      "987942511482496",
      "-4727008292132400886208"
    ],
    [
      "13951748666332005/16",
      "-666805218776552253396811/64"
    ],
    [
      "397195526444269578582656/177129481",
      "184755682663955878491031799200844608/2357416262629"
    ],
    [
      "10083618951832736",
      "996976824896800215156352"
    ],
    [
      "11918542857126748",
      "1286695214692376067191380"
    ],
    [
      "6843911934657346548837664/416445649",
      "17797961083287660114813363226306796416/8498406359143"
    ],
    [
      "2769496506371736",
      "-119782031103604577330648"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=8863/13: 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 12 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 16:04:19",
  "updated_at": "2026-07-01 22:47:36"
}