{
  "id": 185,
  "curve_key": "668466675304964315276964289:-16128439786944158266464185415356070219937",
  "ainvs": [
    "1",
    "0",
    "0",
    "-13926389068853423234936756",
    "18667175679332355979688921260281429136"
  ],
  "rank_lower_bound": 16,
  "naive_height": 185.3010863316041,
  "faltings_height": 13.395464875988132,
  "conductor": "43657243228043341071443078601634748045721626333578493820310",
  "bad_primes": [
    "2",
    "5",
    "7",
    "11",
    "23",
    "29",
    "53",
    "67",
    "127",
    "2749",
    "158647",
    "266621837",
    "124179295151",
    "13053706547096656897"
  ],
  "discriminant": "22324203557420732705257905143620661610548249883929708008721170067485403955200",
  "regulator": "836077628215322.2135091501443589696508094598206988819231525455259700725",
  "points": [
    [
      "152286804214184",
      "-1878728999700747104692"
    ],
    [
      "765300447413032/9",
      "-21151245578527565284156/27"
    ],
    [
      "281550663104079/4",
      "-4717753822362220088699/8"
    ],
    [
      "486836121081064/9",
      "-10716771708385511474876/27"
    ],
    [
      "43233200766510",
      "-283238750412797198806"
    ],
    [
      "1184091393673305064/1071225",
      "2384138221407547739830063268/1108717875"
    ],
    [
      "2991492522774306/2209",
      "157154578672959972375304/103823"
    ],
    [
      "221345423961929704/133225",
      "16527671876532706844540028/48627125"
    ],
    [
      "3240266480360533454/1934881",
      "-550089703964649232827177002/2691419471"
    ],
    [
      "136199984927144/25",
      "1277794623832474021468/125"
    ],
    [
      "11198548828296616/1681",
      "1025845985070022440673452/68921"
    ],
    [
      "20410091612901109370/2337841",
      "84777817445557080735804895966/3574558889"
    ],
    [
      "3669418539293138073664/25170289",
      "222205513477376556361838615217876/126279339913"
    ],
    [
      "209617558513521864/124609",
      "958837106568577566899084/43986977"
    ],
    [
      "2587523431659102184/16129",
      "4161116451083341379626226036/2048383"
    ],
    [
      "1284015905840",
      "1703649433200821444"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-756, 60, 210, 486, 654, -654] with shift t=89/4: 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 16:02:26",
  "updated_at": "2026-07-01 22:39:40"
}