{
  "id": 6520,
  "curve_key": "58988818723290816:-14899964485569283179826560",
  "ainvs": [
    "0",
    "0",
    "0",
    "-1228933723401892",
    "17245329265705188865540"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 115.92680212257366,
  "faltings_height": 7.586805634376886,
  "conductor": "5993455125988316973121681639245851368904",
  "bad_primes": [
    "2",
    "13",
    "23",
    "697593857",
    "2866808023",
    "1252895426640035917"
  ],
  "discriminant": "-9691368991082100639001974237207427696706816768",
  "regulator": "1725874704480.349550669375730202527098856933026432196558301668399029754",
  "points": [
    [
      "-40630567",
      "-10144809571"
    ],
    [
      "425012421616/25",
      "-277077660565223414/125"
    ],
    [
      "-4762179655/121",
      "-90762252819485/1331"
    ],
    [
      "58026840897/841",
      "-12458086215870113/24389"
    ],
    [
      "7681932142/121",
      "-587927160477098/1331"
    ],
    [
      "2401504792/49",
      "-93770288596522/343"
    ],
    [
      "7322324936/289",
      "239334329088842/4913"
    ],
    [
      "93414898801/961",
      "26915699804609447/29791"
    ],
    [
      "936275231336/5329",
      "889203955568132282/389017"
    ],
    [
      "-7418141279/196",
      "268055109073583/2744"
    ],
    [
      "-28077078",
      "-172094190242"
    ],
    [
      "276702201/400",
      "1024360518508301/8000"
    ],
    [
      "-21755610824/1681",
      "-12131370881718998/68921"
    ],
    [
      "-38930999",
      "78001549093"
    ],
    [
      "1038591496354/58081",
      "439898856372958034/13997521"
    ],
    [
      "1224035472449/390625",
      "-28287877700061628207/244140625"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,20,44,87,88,91,102,124,146,168: 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 >= 16: the 16 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 04:56:36",
  "updated_at": "2026-11-04 04:56:36"
}