{
  "id": 166,
  "curve_key": "176988345523030060588628721:-2354176119087252756271839420792499649481",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-3687257198396459595596432",
    "2724740878574131022947561407482810739"
  ],
  "rank_lower_bound": 17,
  "naive_height": 181.31437835328128,
  "faltings_height": 12.823862241473426,
  "conductor": "283351194095753440937593600911468959073927371811953815851270",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "17",
    "37",
    "612349853",
    "628765815428201506421019641699709465077893163"
  ],
  "discriminant": "1153068932160301343602378339411351240373132290030473551029141204930560000",
  "regulator": "58276555418819039.9906295005332111538296892176393442258933331924181239547",
  "points": [
    [
      "23495348691869",
      "-113517683577582019887"
    ],
    [
      "2761907455655033/289",
      "-142419822038161029626487/4913"
    ],
    [
      "8514920884797",
      "-24262997061118929199"
    ],
    [
      "8105002919171097/2401",
      "-630759937393946987000151/117649"
    ],
    [
      "57797791296753/49",
      "-44176752064533223057/343"
    ],
    [
      "1154713200669",
      "-81656470446780783"
    ],
    [
      "4175670052134923/3721",
      "-2520356457515651095617/226981"
    ],
    [
      "7696384561642703/6724",
      "34258403517295224331743/551368"
    ],
    [
      "1197083102477",
      "161916663098376321"
    ],
    [
      "29846231699951433/24649",
      "727314072799652611618793/3869893"
    ],
    [
      "2348554157956317/1681",
      "37768070556822972286521/68921"
    ],
    [
      "1973500432557",
      "1770348455310631841"
    ],
    [
      "3999708899746847",
      "-252954569974946715806049"
    ],
    [
      "79104131395782773/51529",
      "9660347883013118928105483/11697083"
    ],
    [
      "35416531938889181/24025",
      "-2613839719916380864518069/3723875"
    ],
    [
      "6655658584559163/5929",
      "5725043071098605862733/456533"
    ],
    [
      "5297203954988124077/2193361",
      "9133820904479001092726987241/3248367641"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=3303/2: 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 16:00:51",
  "updated_at": "2026-07-01 22:35:42"
}