{
  "id": 2772,
  "curve_key": "136999026458641091728:271728717854921253514889485120",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-2854146384555022744",
    "-314500829899221174494262656"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    4,
    2
  ],
  "naive_height": 139.09951648066317,
  "faltings_height": 9.658098911231493,
  "conductor": "198425172913968",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "13",
    "43",
    "53",
    "73",
    "103",
    "241"
  ],
  "discriminant": "1445290328176598582372191974828917288861611699013105501184",
  "regulator": "3417.660717931345791056081138496159216584618200",
  "points": [
    [
      "1779892412",
      "15625462894860"
    ],
    [
      "-600024131591/1600",
      "-1697033795746381677/64000"
    ],
    [
      "182911948440",
      "-78224820918092128"
    ],
    [
      "41430136437308229/2553604",
      "-8386697639202420897265445/4080659192"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous to curve#1635 via the rational 2-torsion point x=3262736849. Independent witness points are the images of that curve's witnesses under the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.",
  "created_at": "2026-09-26 20:21:45",
  "updated_at": "2026-09-26 20:21:45"
}