{
  "id": 188,
  "curve_key": "2408646883420091075699352961:-119928043257248834891063841588808215302209",
  "ainvs": [
    "1",
    "0",
    "0",
    "-50180143404585230743736520",
    "138805605621811599445299434551151234112"
  ],
  "rank_lower_bound": 16,
  "naive_height": 189.17542110034245,
  "faltings_height": 13.645381835671836,
  "conductor": "3668941577753100929827306465184507435413975438465768915789448790",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "19",
    "307",
    "25219",
    "1733539",
    "159637111",
    "5024333263",
    "22802691139247",
    "1542477233890883"
  ],
  "discriminant": "-236561593535490259936412512778599347267021742231315444284836282901094195200000",
  "regulator": "136960321369372805.8656134456998812155236881189062078900842549168588408",
  "points": [
    [
      "825716257841424",
      "23726316375265516782648"
    ],
    [
      "5525622329342214",
      "-410744203431915461922282"
    ],
    [
      "91443162077961936/49",
      "-27651820816198881418299576/343"
    ],
    [
      "108061543553424",
      "-1120974193470845681352"
    ],
    [
      "56294831605250064/2401",
      "-12807841415343343821981768/117649"
    ],
    [
      "152843981101086/49",
      "1218992269125288065724/343"
    ],
    [
      "216195502811586/49",
      "622611228485546670924/343"
    ],
    [
      "78123844306943544/17161",
      "4879495862885822492127528/2248091"
    ],
    [
      "223153900113936/49",
      "746148161760630044424/343"
    ],
    [
      "1026041124226967808/165649",
      "546182001320023067322019704/67419143"
    ],
    [
      "783388313858485297374/111746041",
      "13548940718902018465047652299948/1181267399411"
    ],
    [
      "39614166327566417136/4870849",
      "176194365587628010721239681224/10749963743"
    ],
    [
      "20346720026928",
      "86839628682789329976"
    ],
    [
      "3460337674856336/2809",
      "1322068898159854927638616/148877"
    ],
    [
      "139553690836715272193916304/2627579402361",
      "1634628281522536737464060336430216707288/4259256287218536141"
    ],
    [
      "70352264711064",
      "587207690274157016568"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=1889/6: 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 16:02:42",
  "updated_at": "2026-07-01 22:40:38"
}