{
  "id": 1115,
  "curve_key": "1800644368:-76408541074880",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-37513424",
    "88448315904"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    4,
    2
  ],
  "naive_height": 63.93423126003121,
  "faltings_height": 2.8181402278888807,
  "conductor": "1092624",
  "bad_primes": [
    "2",
    "3",
    "13",
    "17",
    "103"
  ],
  "discriminant": "202645005149343744",
  "regulator": "49.423766019401803730003747642111266632703600",
  "points": [
    [
      "-3055",
      "417768"
    ],
    [
      "118456/9",
      "36967112/27"
    ],
    [
      "183906/25",
      "57352254/125"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "y^2 = x(x+1)(x+103^2), from the Z/2Z x Z/4Z family y^2 = x(x+r^2)(x+s^2) with (r, s) = (1, 103); the point (rs, rs(r+s)) has order 4. Rank exactly 3 (PARI ellrank bounds agree). No curve with this torsion and rank 3 exists in Cremona's tables (complete to conductor 500000; all 1737 such curves scanned) or in LMFDB; within the family this is the smallest conductor for coprime r < s <= 450.",
  "created_at": "2026-09-23 20:49:49",
  "updated_at": "2026-09-23 20:49:49"
}