{
  "id": 6157,
  "curve_key": "787521416171122926241:-26655774468477925442744613486961",
  "ainvs": [
    "1",
    "0",
    "0",
    "-16406696170231727630",
    "30851590818630374211212880900"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    7
  ],
  "family": null,
  "families": [],
  "naive_height": 144.7211171899072,
  "faltings_height": 10.058754933371297,
  "conductor": "324387832897590",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "61",
    "135417197"
  ],
  "discriminant": "-128540173864967991026733414541042638776771211920670720000000",
  "regulator": "704.4249089445045526693764378107691553776384073",
  "points": [
    [
      "-3613980740",
      "207229676507170"
    ],
    [
      "5170751260",
      "290282289781570"
    ],
    [
      "14944268860",
      "1767246495975970"
    ],
    [
      "1088832839622940/316969",
      "21875786221257322852790/178453547"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Specialization at u=29/17, giving t=187/427, of Leopoldo Kulesz’s family in “Families of elliptic curves of high rank with nontrivial torsion group over Q”, Acta Arithmetica 108 (2003), Theorem 2.6, also cited on curve#1213 (David Renshaw), curve#4550 and curve#4552. https://www.impan.pl/shop/en/publication/transaction/download/product/82638 \r\n\r\nHere t=-2*(u-3)/(u^2+3), b=t^2*(t-1), c=t*(t-1), and the source equation is y^2+(1-c)*x*y-b*y=x^3-b*x^2.\r\n\r\nSelected by a bounded rational-parameter search. PARI/GP returned rank bounds [4,4], four independent points and torsion Z/7Z. All four points were checked exactly on the displayed global minimal model.",
  "created_at": "2026-10-29 19:17:37",
  "updated_at": "2026-10-29 19:17:37"
}