{
  "id": 1210,
  "curve_key": "10729028881:-1651786619903225",
  "ainvs": [
    "1",
    "1",
    "1",
    "-223521435",
    "1911696935401"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    5
  ],
  "naive_height": 70.081267793919,
  "faltings_height": 3.868756995763802,
  "conductor": "2462003614",
  "bad_primes": [
    "2",
    "7",
    "17",
    "43",
    "240571"
  ],
  "discriminant": "-864212044892876703988579328",
  "regulator": "97.51278782964378544263322840471627982471037234",
  "points": [
    [
      "-10397",
      "1769254"
    ],
    [
      "635",
      "1330102"
    ],
    [
      "10547",
      "847718"
    ],
    [
      "228793",
      "109097734"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found by an independent rational-parameter search in the Tate family y^2+(1-t)xy-ty=x^3-tx^2 at t=119/172, whose point (0,0) has order 5. Submitted in the global minimal model [1,1,1,-223521435,1911696935401]. PARI/GP ellrank returns bounds [4,4], proving rank exactly 4, and elltors returns invariant factors [5], proving torsion exactly Z/5Z. The four supplied rational points were found by PARI and independently checked by the ICARM exact independence verifier.",
  "created_at": "2026-09-24 13:01:15",
  "updated_at": "2026-09-24 13:01:15"
}