{
  "id": 6232,
  "curve_key": "1259513977977:-1413537252534648909",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-26239874541",
    "1636045046698553"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    3
  ],
  "family": null,
  "families": [],
  "naive_height": 83.58525385302926,
  "faltings_height": 4.581599749866864,
  "conductor": "1392454680029442",
  "bad_primes": [
    "2",
    "3",
    "37",
    "61",
    "89",
    "103",
    "233",
    "1783"
  ],
  "discriminant": "-14755328331716680838495722116",
  "regulator": "14165.35669738120207828170506903632271019595591251202",
  "points": [
    [
      "-2549726/25",
      "-7121202967/125"
    ],
    [
      "93487",
      "57253"
    ],
    [
      "2339756/25",
      "-19360569/125"
    ],
    [
      "77371",
      "8267881"
    ],
    [
      "114331204/1225",
      "4179680147/42875"
    ],
    [
      "2345986/25",
      "18646169/125"
    ],
    [
      "21084049/256",
      "-23907207983/4096"
    ]
  ],
  "submitter": "Lewis Combes",
  "history": [],
  "commentary": "E has torsion Z/3Z and rank exactly 7. It is isomorphic to the Tate form y^2 + 1047xy - 2301807y = x^3 (a1 = 1047, a3 = -2301807), in which (0,0) is a point of order 3. Every curve with a rational point of order 3 has such a form y^2 + a1 xy + a3 y = x^3, and (a1, a3) and (a1/u, a3/u^3) give isomorphic curves (x -> u^2 x, y -> u^3 y), so the curve is determined by t = a3/a1^3: it is the member t = -767269/382576941 (in lowest terms) of the one-parameter Z/3 family y^2 + xy + ty = x^3. The submitted a-invariants are the global minimal model of this curve (the Tate form above is already minimal; the two differ by a translation), on which the point (0,0) of the Tate form is (91351, -1196579).\r\n\r\nRank exactly 7: the seven points listed are independent (regulator 14165.357), and the 2-Selmer group has rank 7 (E has no rational 2-torsion; root number -1), so the rank is at most 7. The Selmer computation assumes GRH for the class groups of the cubic fields involved. Confirmed in PARI/GP (ellrank [7, 7]). It beats the previous T[3] rank-7 records in discriminant (#1661, https://elliptic-rank.icarm.cloud/curve/1661).\r\n\r\nSearch scoped and carried out by Claude code Opus 5.5",
  "created_at": "2026-10-31 10:57:56",
  "updated_at": "2026-10-31 10:57:56"
}