{
  "id": 198,
  "curve_key": "72945936872713395584521:-19759029617750051108484067702196069",
  "ainvs": [
    "1",
    "1",
    "0",
    "-1519707018181529074677",
    "22869247242133051599466107911241"
  ],
  "rank_lower_bound": 14,
  "naive_height": 157.93783730300783,
  "faltings_height": 10.97539098710795,
  "conductor": "219931200577530279341714779925105066004870",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "17",
    "31",
    "67",
    "163",
    "2671",
    "19213",
    "272760893775113648578459"
  ],
  "discriminant": "-1311295550412181298956751711791068502351802457869713490554687500",
  "regulator": "57498099076.9160905642634205944578651803129223187865761115602274920",
  "points": [
    [
      "29428257775715296/235225",
      "-4827910858317241171052311/114084125"
    ],
    [
      "388726833316031142/13256881",
      "90552776990454414052053489/48268303721"
    ],
    [
      "7911228532252973/258064",
      "295806306648889730642983/131096512"
    ],
    [
      "121528141873272/3721",
      "644932695710007776079/226981"
    ],
    [
      "12410076961752/361",
      "23007869174866244481/6859"
    ],
    [
      "35203694052",
      "3605240641688769"
    ],
    [
      "47187126232",
      "7498420893276259"
    ],
    [
      "84857410752",
      "22471059109025379"
    ],
    [
      "109029826357",
      "33959748881502634"
    ],
    [
      "1070008775432",
      "1106105206155389859"
    ],
    [
      "1212557108788/9",
      "-1284613342747699957/27"
    ],
    [
      "37891831182",
      "-4437305361723291"
    ],
    [
      "33961043607",
      "-3229175523309741"
    ],
    [
      "28478113232",
      "-1639127433627741"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-756, 60, 210, 486, 654, -654] with shift t=1788/5: 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 14 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 16:03:35",
  "updated_at": "2026-07-01 22:44:38"
}