{
  "id": 6519,
  "curve_key": "320361736739118256:-140534053254290602719301400",
  "ainvs": [
    "0",
    "-1",
    "1",
    "-6674202848731630",
    "162655156454237295317142"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 120.92468153868707,
  "faltings_height": 8.091511500929686,
  "conductor": "133299129597562732119902707030702830923252701821",
  "bad_primes": [
    "3",
    "19",
    "20051",
    "50363",
    "16062394022628251",
    "144176524948432809631"
  ],
  "discriminant": "7598050387061075730834454300750061362625404003797",
  "regulator": "289228768909796.132825990042882451536063144491613105588244656028195671145",
  "points": [
    [
      "67130623210/169",
      "-17044473347809454/2197"
    ],
    [
      "2558519358640",
      "-4092446972518509107"
    ],
    [
      "423545399869/900",
      "-271678031936179253/27000"
    ],
    [
      "10767162129549/625",
      "-35330273748906431157/15625"
    ],
    [
      "-130519816780/22201",
      "1485595594224188369/3307949"
    ],
    [
      "579992286377/23104",
      "-367130259446973685/3511808"
    ],
    [
      "2701153030221/24025",
      "3399707111296793281/3723875"
    ],
    [
      "102696684549/676",
      "28637889404042745/17576"
    ],
    [
      "34238904821041/30625",
      "199821682571917682536/5359375"
    ],
    [
      "1332398610862642/4313929",
      "-47042045043313216906430/8960030533"
    ],
    [
      "42466523433/121",
      "-8527767502633371/1331"
    ],
    [
      "15648407479053676/53246209",
      "-1886879949710252035357883/388537587073"
    ],
    [
      "786005229124/729",
      "-694890240112939906/19683"
    ],
    [
      "16507695908094/175561",
      "-44528138582075762570/73560059"
    ],
    [
      "-1832236067/36",
      "-131477276264005/216"
    ],
    [
      "4673003443821454/43678881",
      "-236845320885056808024659/288673724529"
    ],
    [
      "3017550840270/16129",
      "-4787471514194022402/2048383"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,13,60,75,149,152,155,156,175,191: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 17: the 17 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.",
  "created_at": "2026-11-04 04:56:29",
  "updated_at": "2026-11-04 04:56:29"
}