{
  "id": 165,
  "curve_key": "1304157770020534929103842697:-46841312725661806886923237084796777780389",
  "ainvs": [
    "1",
    "0",
    "1",
    "-27169953542094477689663390",
    "54214482321365567882254868382411020852"
  ],
  "rank_lower_bound": 17,
  "naive_height": 187.3060648689139,
  "faltings_height": 13.448120467938107,
  "conductor": "5055747593755543039707388407261319968257026830606998385654",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "23",
    "43",
    "127",
    "283",
    "78439",
    "212867",
    "29444935757",
    "529852873647245252778716011"
  ],
  "discriminant": "13911346942980486161686871517934679432553631814179561190438456955749228983884",
  "regulator": "12547921251179166.6765132651478498300270414723508308527523598212195212239",
  "points": [
    [
      "145057606823895",
      "1745958976309376327161"
    ],
    [
      "522683745173552301/6889",
      "377014652656645941021061133/571787"
    ],
    [
      "1583807608055806377/65536",
      "1950214090754745481698364267/16777216"
    ],
    [
      "25355780823667965/4489",
      "2706170730021416266520329/300763"
    ],
    [
      "356788907950971291/157609",
      "-129887064946403282327521283/62570773"
    ],
    [
      "401598226891907958/142129",
      "-2799121654846465741600120/53582633"
    ],
    [
      "1431224065644819/529",
      "8685208265999749724357/12167"
    ],
    [
      "1908207155420615124/811801",
      "1336094215002135384439721779/731432701"
    ],
    [
      "9576692532879",
      "-25929211779693034199"
    ],
    [
      "1987724382604458/169",
      "-81072829838071536279911/2197"
    ],
    [
      "31966715613507",
      "-178470005016992136695"
    ],
    [
      "1219124517544603/9",
      "-42535846884691295629151/27"
    ],
    [
      "732380383456113441843/141300769",
      "12185573817003989870271410080045/1679642241103"
    ],
    [
      "1180741943486364",
      "-40572172937314419910964"
    ],
    [
      "1072640820364224006741/2765569",
      "35127112227364754330025618978517/4599141247"
    ],
    [
      "6885162389871",
      "13911834614960687977"
    ],
    [
      "2562767237796",
      "-1189915762220229473"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=2906/3: 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 16:00:45",
  "updated_at": "2026-07-01 22:36:40"
}