{
  "id": 6231,
  "curve_key": "597694431897:-462157195243111101",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-12451967331",
    "534907274152841"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    3
  ],
  "family": null,
  "families": [],
  "naive_height": 81.34936295512935,
  "faltings_height": 4.4903297491567375,
  "conductor": "1371948579064818",
  "bad_primes": [
    "2",
    "3",
    "53",
    "61",
    "67",
    "89",
    "271",
    "1621"
  ],
  "discriminant": "-40353450288992772493645578276",
  "regulator": "16059.55290687260014734553967487654601500983745667776",
  "points": [
    [
      "137198924/1369",
      "865313390355/50653"
    ],
    [
      "-58272653/1156",
      "1265152516933/39304"
    ],
    [
      "64784",
      "302515"
    ],
    [
      "117768965/1849",
      "-37246159722/79507"
    ],
    [
      "17849600129/4096",
      "-2384546069909855/262144"
    ],
    [
      "-50007176/961",
      "962410037941/29791"
    ],
    [
      "15363407/361",
      "-62278057026/6859"
    ]
  ],
  "submitter": "Lewis Combes",
  "history": [],
  "commentary": "E has torsion Z/3Z and rank exactly 7. It is isomorphic to the Tate form y^2 + 849xy - 3834921y = x^3 (a1 = 849, a3 = -3834921), in which (0,0) is a point of order 3.\r\nEvery 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 = -1278307/203986683 (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 (60067, -1947494).\r\n\r\nIt was found by a screen of the discriminant cusps of the Z/3 family: all minimal models y^2 + a1 xy + a3 y = x^3 with |Delta| = |a3|^3 |a1^3 - 27 a3| <= 1.18 x 10^30 (ten times the previous T[3] rank-7 discriminant record). Most of these have a3 small against a1^3 (|t| small) and lie outside the boxes searched before. A Mestre-Nagao screen in C over 3.6 x 10^11 models left 1,745 new candidates for Magma (2-Selmer rank, 2-descent with point search, RankBounds). This is the discriminant-first search of the integral Tate X_1(3) family that jesper-petersen used at rank 3 (#3759, https://elliptic-rank.icarm.cloud/curve/3759), carried to rank 7.\r\n\r\nSearch scoped and carried out by Claude code Opus 5.5",
  "created_at": "2026-10-31 10:54:45",
  "updated_at": "2026-10-31 10:54:45"
}