{
  "id": 6344,
  "curve_key": "649743063472226518711921:-532857778220034780338224458020488969",
  "ainvs": [
    "1",
    "0",
    "0",
    "-13536313822338052473165",
    "616733539605393710350713099817617"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    5
  ],
  "family": null,
  "families": [],
  "naive_height": 164.52712524951076,
  "faltings_height": 11.599368594178896,
  "conductor": "2302503118138763391810",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "29",
    "31",
    "71",
    "131",
    "839",
    "7321",
    "213481"
  ],
  "discriminant": "-5577517444193886430538050972866251518998430542653867962977484800000",
  "regulator": "150048.7655530321083135437497720889766023995202985743",
  "points": [
    [
      "551087389498866/1849",
      "12073063687780290354861/79507"
    ],
    [
      "-1557115866714396/83521",
      "708928686994438244674797/24137569"
    ],
    [
      "-106109868143559952254/797893009",
      "-181590015290431840080675464391/22538083825223"
    ],
    [
      "103420053697963434/1635841",
      "7727474673810190701048057/2092240639"
    ],
    [
      "43873563169282146/8254129",
      "-553577821474555518470380329/23714112617"
    ],
    [
      "12374141738968997394/192071881",
      "-9243589092261568159107960123/2661924198779"
    ],
    [
      "1378359017447676/26569",
      "31858044357770286867591/4330747"
    ]
  ],
  "submitter": "Lewis Combes",
  "history": [],
  "commentary": "E has torsion Z/5Z and rank exactly 7. It is isomorphic to the Tate normal form y^2 + (1 - t)xy - ty = x^3 - tx^2 at t = 132965/633888, in which (0,0) is a point of order 5.\r\nEvery curve with a rational point P of order 5 can be put in this form (P moved to (0,0) and its tangent to y = 0); t is determined by (E, P), and P -> 2P replaces t by -1/t.\r\nThe submitted a-invariants are the global minimal model of this curve, on which the point (0,0) of the Tate normal form is (-7184651646, -26713595432910657).\r\n\r\nIt was found by a screen of the discriminant region of this family: all t = a/b with |Delta_min| <= 4.93 x 10^76 (the T[5] rank-8 discriminant record, #1917, https://elliptic-rank.icarm.cloud/curve/1917), most of them in the cusp where |a| is small against b, outside the boxes searched before. A Mestre-Nagao screen in C over 3.4 x 10^13 parameters left 335,529 new candidates for Magma (2-Selmer rank, 2-descent with point search, RankBounds). The family and kind of search are those of hdeping's Mestre-Nagao sieve over the Tate normal form (#1254, https://elliptic-rank.icarm.cloud/curve/1254) and David Renshaw's rational-parameter searches (#1185, https://elliptic-rank.icarm.cloud/curve/1185), carried to the whole region below a discriminant bound.\r\n\r\nSearch scoped and carried out by Claude code Opus 5.5",
  "created_at": "2026-11-01 03:15:42",
  "updated_at": "2026-11-01 03:15:42"
}