{
  "id": 1217,
  "curve_key": "367311529:-299420282470933",
  "ainvs": [
    "1",
    "0",
    "1",
    "-7652324",
    "346550615166"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    3
  ],
  "naive_height": 66.66573865868693,
  "faltings_height": 3.6132202392665893,
  "conductor": "534549753010",
  "bad_primes": [
    "2",
    "5",
    "7",
    "13",
    "83",
    "499",
    "1091"
  ],
  "discriminant": "-51853558279225827903114700",
  "regulator": "4645.27913151823259864686293512325574245596803013",
  "points": [
    [
      "-1077",
      "595133"
    ],
    [
      "-571",
      "592513"
    ],
    [
      "55991/25",
      "72816221/125"
    ],
    [
      "225637/9",
      "107367311/27"
    ],
    [
      "17050584242/12769",
      "2225461474939871/1442897"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found by an independent search in the family y^2+a*x*y+b*y=x^3 with b=k^2-(a+1)*k, which contains the rational point (k,k) and the point (0,0) of order 3. Parameters a=-13, k=1079 give b=1177189, hence y^2-13*x*y+1177189*y=x^3. Submitted in its global minimal model [1,0,1,-7652324,346550615166]. PARI/GP ellrank returns bounds [5,5], proving rank exactly 5, and elltors returns invariant factors [3], proving torsion exactly Z/3Z. The five supplied rational points were computed by PARI and independently checked by the ICARM exact independence verifier.",
  "created_at": "2026-09-24 14:22:44",
  "updated_at": "2026-09-24 14:22:44"
}