{
  "id": 1213,
  "curve_key": "151902702823412904609:-1875483561054907893754207476081",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-3164639642154435513",
    "2170698566826932935717089753"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    7
  ],
  "naive_height": 139.4128386304455,
  "faltings_height": 9.412209349796704,
  "conductor": "9497014859878",
  "bad_primes": [
    "2",
    "7",
    "13",
    "17",
    "19",
    "37",
    "1399",
    "3121"
  ],
  "discriminant": "-7158642232748587940865724518143897565548574582499180544",
  "regulator": "16862.75030412072859087578818390112635755087246",
  "points": [
    [
      "-1580214169",
      "56795074990028"
    ],
    [
      "1254497139",
      "13226181118668"
    ],
    [
      "-65393725583903721/60109009",
      "30651618287999517029483756/466025146777"
    ],
    [
      "5401567490195658687/5348435689",
      "849629272887962506618871868/391147147243637"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found by an independent specialization search in the rank-at-least-1 family from Leopoldo Kulesz, Families of elliptic curves of high rank with nontrivial torsion group over Q, Acta Arithmetica 108 (2003), Theorem 2.6: https://www.impan.pl/shop/en/publication/transaction/download/product/82638 . At u=13/17, the Tate parameter is t=-2*(u-3)/(u^2+3)=323/259. The curve y^2+(1-c)xy-by=x^3-bx^2, where b=t^2*(t-1) and c=t*(t-1), is submitted in its global minimal model. PARI/GP ellrank returns bounds [4,4], proving rank exactly 4, and elltors returns invariant factors [7], proving torsion exactly Z/7Z. The four supplied rational points were computed by PARI and independently checked by the ICARM exact independence verifier.",
  "created_at": "2026-09-24 13:44:21",
  "updated_at": "2026-09-24 13:44:21"
}