{
  "id": 217,
  "curve_key": "258832074689910424162425:-131680663259199131919558611354338125",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-5392334889373133836717",
    "152408175069865597518128713267441"
  ],
  "rank_lower_bound": 16,
  "naive_height": 161.7313993320824,
  "faltings_height": 11.110527881500825,
  "conductor": "3865572244758359427703312649778700630767441782587550",
  "bad_primes": [
    "2",
    "5",
    "11",
    "43",
    "47",
    "113233229",
    "2477679059415479",
    "12395551355742150820531"
  ],
  "discriminant": "237289500362812252947418258536952148541504393969458434158062500",
  "regulator": "1035675008217125.210650282138077450989277944286150483848078016626924898",
  "points": [
    [
      "-11886849988520801543020205/219020050054276",
      "-54739616447895393532106980635723997109/3241349121289548217976"
    ],
    [
      "168878841911/4",
      "371469986103239/8"
    ],
    [
      "93228044365251/4",
      "-900155943328475632151/8"
    ],
    [
      "7162568606295437/169744",
      "4001827171142801549277/69934528"
    ],
    [
      "-25775659536156/361",
      "-90325109045635789993/6859"
    ],
    [
      "185255057136930686/4355569",
      "-218161975946970898534041/9090072503"
    ],
    [
      "10529263292731/49",
      "32387293583565702279/343"
    ],
    [
      "864197995016095/20449",
      "-66655557619693066292/2924207"
    ],
    [
      "-42757882743809/529",
      "94403070231401230891/12167"
    ],
    [
      "194866459365465611/4566769",
      "-861267473996490181449116/9759185353"
    ],
    [
      "-8287086919/64",
      "6335292054507076401/512"
    ],
    [
      "4898229429/16",
      "785813137131910117/64"
    ],
    [
      "363416598",
      "12265747209548093"
    ],
    [
      "-576940395",
      "12470727566797076"
    ],
    [
      "700086904",
      "12191530118261073"
    ],
    [
      "-4958351906",
      "13379963865786753"
    ]
  ],
  "submitter": "Edgar Costa",
  "commentary": "Specialization at T = -2068 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (-44, -60, -6, 110, 94, -94) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Sixteen independent rational points were obtained via 2-descent covers and a direct minimal-model x = n/q^2 sieve (the extra generators are high-height on the minimal model but small on the covers); rank >= 16 follows from the positive-definiteness of their 16x16 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).",
  "created_at": "2026-06-25 21:53:06",
  "updated_at": "2026-07-01 22:38:38"
}