{
  "id": 190,
  "curve_key": "22216343063906422078063211809:-3296420228104962701344386219541803645673777",
  "ainvs": [
    "1",
    "0",
    "0",
    "-462840480498050459959650246",
    "3815301189936260852812257361005424248676"
  ],
  "rank_lower_bound": 16,
  "naive_height": 195.8118771087723,
  "faltings_height": 14.149337069918305,
  "conductor": "105796999633828233559899461972111361932925512955353458023082540370",
  "bad_primes": [
    "2",
    "5",
    "13",
    "17",
    "31",
    "59",
    "89",
    "157",
    "10687",
    "4882196499566433409319",
    "2761616040077202452279063069"
  ],
  "discriminant": "57200821486488614373193385715411199336388466289969673621065223447387681934540800",
  "regulator": "214216026004757241.3332241685125651774174884966523042487861989266602557",
  "points": [
    [
      "469312363122379396/2401",
      "-319638192435662415699200342/117649"
    ],
    [
      "75846233912356206459580/680844649",
      "-20524335079905029521908270923524558/17765279426357"
    ],
    [
      "1407409071959436/25",
      "-49399211055052979742974/125"
    ],
    [
      "28213970481676",
      "-114960214643784858518"
    ],
    [
      "8381669212173592671196/573842025",
      "-179793433897362095540192541040034/13746385708875"
    ],
    [
      "3005595537603714604/227529",
      "273724828703003818689318386/108531333"
    ],
    [
      "6603562616910707537996/497334601",
      "36247012241943409079347316562962/11091058936901"
    ],
    [
      "8775236465006116/625",
      "143764105112299072042714/15625"
    ],
    [
      "70619085957476",
      "568604587009163644042"
    ],
    [
      "72904017699180",
      "597960596232290981546"
    ],
    [
      "1201852798686484/9",
      "41155073865803806865014/27"
    ],
    [
      "442092042524249452/1089",
      "293542342404627768229271914/35937"
    ],
    [
      "1193902467929398508/22801",
      "1208195596369186292976575590/3442951"
    ],
    [
      "470290282919339747660/30791401",
      "-3003906506723684743914116368974/170861484149"
    ],
    [
      "3239968495326088/81",
      "-161864274616498478591270/729"
    ],
    [
      "841154609703266",
      "-24387823893121165826928"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=1069/4: 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 16:02:53",
  "updated_at": "2026-07-01 22:40:39"
}