{
  "id": 6502,
  "curve_key": "17046448232450621964096:-2185721448556755261725568308338560",
  "ainvs": [
    "0",
    "0",
    "0",
    "-355134338176054624252",
    "2529770195088837108478667023540"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 153.57068646027497,
  "faltings_height": 10.688682557469038,
  "conductor": "17781650931748464731843618509615062726355017661111990269304",
  "bad_primes": [
    "2",
    "179",
    "18289",
    "89917",
    "7550875227399260691431851949878486369563079369"
  ],
  "discriminant": "101853296537055205984000246823075079296561541162849480262573312",
  "regulator": "1362623900856756772.029354389510767018000635580062221453325548828882729961697",
  "points": [
    [
      "920374857820",
      "-882788706267899030"
    ],
    [
      "-1517999323286/81",
      "1176204963351356126/729"
    ],
    [
      "1912442196734/25",
      "-2570929267699455602/125"
    ],
    [
      "11009488660796/361",
      "-30723401364246529138/6859"
    ],
    [
      "720480976786/25",
      "-503594047732456534/125"
    ],
    [
      "2829954255258/361",
      "-3271839682779740954/6859"
    ],
    [
      "88623750026554/9409",
      "-130314193242982898674/912673"
    ],
    [
      "547314005478316/40401",
      "3676176938608650803798/8120601"
    ],
    [
      "15752485256513/1156",
      "18467234975775836801/39304"
    ],
    [
      "18002472500734/2025",
      "24990244733563588898/91125"
    ],
    [
      "-18327019570588/3481",
      "-423577480467713871802/205379"
    ],
    [
      "42005402852763098/6806881",
      "13445808112974383599264174/17759152529"
    ],
    [
      "442806097681/64",
      "-325375369949912057/512"
    ],
    [
      "211589291222665882/22934521",
      "21587059047022526252967478/109833421069"
    ],
    [
      "9645406286",
      "41287028073818"
    ],
    [
      "1506904390984186/156025",
      "-1681563532602358367434/61629875"
    ],
    [
      "-3380587014",
      "1921379226651982"
    ],
    [
      "268857435935280754/33327529",
      "-83832710209474485246826586/192399824917"
    ],
    [
      "77501353732198/16129",
      "1979909111045043844954/2048383"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,10,19,20,76,97,134,136,169: 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 >= 19: the 19 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 04:54:28",
  "updated_at": "2026-11-04 04:54:28"
}