{
  "id": 159,
  "curve_key": "63443960979800259369:-213981041085997763351391274197",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-1321749187079172070",
    "247663242328119893241310696"
  ],
  "rank_lower_bound": 17,
  "naive_height": 136.79006605770067,
  "faltings_height": 9.455500750508849,
  "conductor": "84815582549450028799125905482004857770488310327228890",
  "bad_primes": [
    "2",
    "5",
    "11",
    "13",
    "10541939988843133587887",
    "5626250600098076816239435529"
  ],
  "discriminant": "121286283045713541182750044839266946611798283767937312700",
  "regulator": "3039850234600066.55872027922921653126582444319920232392965022269415518269",
  "points": [
    [
      "17949979918713078398541/6617884875625",
      "-2194610111437610055414042537451414/17024674289667203125"
    ],
    [
      "1069553418",
      "7581789774136"
    ],
    [
      "1914845121457/37249",
      "-96411962079340334623/7189057"
    ],
    [
      "212468732619099/113569",
      "2516334388237502223941/38272753"
    ],
    [
      "2552123861358/1681",
      "-2875329490394494594/68921"
    ],
    [
      "209259772307129/135424",
      "2169286183762609158459/49836032"
    ],
    [
      "4700687044159/1225",
      "9746723920117144932/42875"
    ],
    [
      "-4095591542092/3481",
      "-2709694728931345656/205379"
    ],
    [
      "4122774202837/3721",
      "-2717812918567624915/226981"
    ],
    [
      "-1855427204865/2401",
      "-3343323765033902131/117649"
    ],
    [
      "-10901289782/9",
      "-228387183419464/27"
    ],
    [
      "61860660194/25",
      "13765447649461718/125"
    ],
    [
      "60291876902082/16129",
      "446623168719117198098/2048383"
    ],
    [
      "13915417197647147/4583881",
      "1527058322512021184486745/9814089221"
    ],
    [
      "2959589615",
      "149194293187741"
    ],
    [
      "1298436411595/484",
      "1347253441578324783/10648"
    ],
    [
      "168529858382/49",
      "65428320462144828/343"
    ]
  ],
  "submitter": "Edgar Costa",
  "commentary": "Rank ≥ 17. This curve is the specialization at T = 2454 of the Mestre/Fermigier family y² = r(x,T), where r is the degree-≤4 remainder in p₆(x−T)·p₆(x+T) = g(x)² − r (g monic of degree 6), with p₆(x) = ∏ᵢ(x−aᵢ) on the sextuple (a) = (−1146, −2304, −654, 3054, 2880, −1830). It was found by Alexey Pozdnyakov via a Mestre–Nagao sieve of this family for small-conductor, high-rank specializations. The rank lower bound is witnessed by the 17 listed independent rational points, whose 17×17 Néron–Tate height-pairing matrix is positive-definite; the conductor N = 2·5·11·13·10541939988843133587887·5626250600098076816239435529 is by Tate's algorithm. Proven lower bound on the Mordell–Weil rank (no exact-rank, Selmer, or BSD claim). Curve and search by Alexey Pozdnyakov.",
  "created_at": "2026-06-25 00:14:13",
  "updated_at": "2026-07-01 22:33:42"
}