{
  "id": 172,
  "curve_key": "7004746952893110001279041:-19171511021279916944243289562775320161",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-145932228185273125026647",
    "22189248867258609149920496015950319"
  ],
  "rank_lower_bound": 16,
  "naive_height": 171.6929774478431,
  "faltings_height": 12.226738906585615,
  "conductor": "25711696321134843123753387143512024261228018943567970",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "19",
    "37",
    "904594283708550341",
    "26425921171690203209101347263"
  ],
  "discriminant": "-13801249682874560225185953107990803809947314431823775922758046720000000",
  "regulator": "492472590511832.3165935628802829899244997577628812687516700439862003778",
  "points": [
    [
      "77386157874918547/323761",
      "5719519573656468731547374/184220009"
    ],
    [
      "-7462742259149/100",
      "180732369761751418069/1000"
    ],
    [
      "72287956542747867/546121",
      "29080163520921100036204234/403583419"
    ],
    [
      "20575728680277/121",
      "63706932078410060606/1331"
    ],
    [
      "4080864960914165907/22363441",
      "4277533349985510929005804654/105756712489"
    ],
    [
      "6220070339342547/28561",
      "131038128988252422570574/4826809"
    ],
    [
      "393063052397707/1681",
      "2010382117608086193486/68921"
    ],
    [
      "866141278428643/2209",
      "16497886662035545362978/103823"
    ],
    [
      "293058398233203/529",
      "4060260064355711051762/12167"
    ],
    [
      "2965346037177",
      "5066018568264726586"
    ],
    [
      "26499730120707",
      "-136400935028918075714"
    ],
    [
      "-1934745224044713/11881",
      "264246600850528066689094/1295029"
    ],
    [
      "-1427985823453196933/11485321",
      "7628586664397936536701775734/38923752869"
    ],
    [
      "-840290327883885983/19280881",
      "14284229246454023303849298956/84662348471"
    ],
    [
      "-27104667413",
      "161631633543294526"
    ],
    [
      "6309240191833083/29929",
      "145887234306918928925462/5177717"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=501/10: 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 16:01:22",
  "updated_at": "2026-07-01 22:38:41"
}