{
  "id": 6204,
  "curve_key": "21287136:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-443482",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 50.62084051371074,
  "faltings_height": 2.286643787784052,
  "conductor": "12587282196736",
  "bad_primes": [
    "2",
    "13",
    "37",
    "461"
  ],
  "discriminant": "5582233083172874752",
  "regulator": "2058.85414547788041074898476176565993587980700976",
  "points": [
    [
      "-481",
      "10101"
    ],
    [
      "666",
      "222"
    ],
    [
      "1274",
      "38766"
    ],
    [
      "3146",
      "172458"
    ],
    [
      "-4329/25",
      "1057719/125"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Specialization at (u,v)=(7,3) of the family in Yasutsugu Fujita and Nobuhiro Terai, On the Rank of the Elliptic Curve y^2 = x^3 - n*x (2012), Theorem 3.3: https://www.m-hikari.com/ija/ija-2012/ija-17-20-2012/teraiIJA17-20-2012.pdf . Here n=u^8+6*u^4*v^4+25*v^8=7095712 and the source equation is y^2=x^3-n*x. The three source sections seeded my point search, followed by the minimal-model change [u,r,s,t]=[4,0,0,0]. PARI/GP returned rank bounds [5,5]. All five listed rational points were checked exactly on the displayed global minimal model and independently certified. The rational torsion subgroup is Z/2Z.",
  "created_at": "2026-10-30 16:42:44",
  "updated_at": "2026-10-30 16:42:44"
}