{
  "id": 164,
  "curve_key": "236935387067387700964671009:-3663721025416020578355068837460581107377",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-4936153897237243770097313",
    "4240417853491998596755638983929308017"
  ],
  "rank_lower_bound": 17,
  "naive_height": 182.19859586265878,
  "faltings_height": 13.015054484977728,
  "conductor": "771012142533763749869410487682169741745976568727255733990",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "17",
    "31",
    "37",
    "423061",
    "148356075867152734210357352683148523657123407"
  ],
  "discriminant": "-70418701085637647451617054505242098864199452815445343344812504364494028800",
  "regulator": "11352746991815636.1172596771173279037714821445010842152357302799389226429",
  "points": [
    [
      "79842022211577577/65536",
      "-3137523758908713317967093/16777216"
    ],
    [
      "17605289416521495703/1437601",
      "-72730269278505203939381021988/1723683599"
    ],
    [
      "63242764747672111/42025",
      "-4042271758419273091936556/8615125"
    ],
    [
      "26065785237334003/17161",
      "-1117454965119437110793952/2248091"
    ],
    [
      "1659653179695",
      "-787111906710352892"
    ],
    [
      "1942280703063",
      "-1407192016767310652"
    ],
    [
      "2117518569559508823/1087849",
      "-1607545365000713659257625876/1134626507"
    ],
    [
      "846101499149193063/249001",
      "-642052115317737566574203348/124251499"
    ],
    [
      "80078043136623",
      "-716316208603657433972"
    ],
    [
      "70814865534903",
      "595628366373660638308"
    ],
    [
      "1128262932027343/81",
      "37442838388387048510652/729"
    ],
    [
      "194476369296111/25",
      "2611815033391840449044/125"
    ],
    [
      "3517375303503884887/2331729",
      "1696029492038458545985440364/3560550183"
    ],
    [
      "10677891747607/9",
      "-6277332203626315604/27"
    ],
    [
      "6392201212718727/5041",
      "-50731235352220994624260/357911"
    ],
    [
      "187849311498195823/143641",
      "-8018354400294797441730068/54439939"
    ],
    [
      "93969411133650535/68121",
      "-4213817666448885805296844/17779581"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=1383/4: 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 17 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 16:00:40",
  "updated_at": "2026-07-01 22:35:42"
}