{
  "id": 6515,
  "curve_key": "2477409373798874361:-3812233531934691315847519245",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-51612695287476549",
    "4412307341531288770766729"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 127.06123524224414,
  "faltings_height": 8.489669188542532,
  "conductor": "422185255072396715012517685057928778442049058",
  "bad_primes": [
    "2",
    "3",
    "11",
    "47",
    "1484303",
    "8670331",
    "134820667",
    "2377018785299191673"
  ],
  "discriminant": "388956742386668659160742468137759436005989144841052",
  "regulator": "40178341575823.9795606735027207739965759477295087676473977783988808276441",
  "points": [
    [
      "9748802407",
      "-962303751395147"
    ],
    [
      "19169251984",
      "-2653865763418877"
    ],
    [
      "-2025192590783/12544",
      "-4104797099339218745/1404928"
    ],
    [
      "119675069332/1681",
      "72239455222577843/68921"
    ],
    [
      "2073984897979/13924",
      "-280988587709656385/1643032"
    ],
    [
      "571181177257/3721",
      "-74127092318831612/226981"
    ],
    [
      "164599772",
      "-613584580967"
    ],
    [
      "447199992848/1369",
      "239777989626726169/50653"
    ],
    [
      "9780635714/25",
      "830073718864577/125"
    ],
    [
      "1556233501/25",
      "-150038578647964/125"
    ],
    [
      "106013020",
      "-363569948903"
    ],
    [
      "62040661048/361",
      "-5393091673645385/6859"
    ],
    [
      "-4148736907/16",
      "38496153691845/64"
    ],
    [
      "212754843905620/597529",
      "-2579032931665666518979/461889917"
    ],
    [
      "9500942109277/26896",
      "-24264894066213290725/4410944"
    ],
    [
      "7473798148",
      "-645826882945295"
    ],
    [
      "4217332450429/44944",
      "-5992125157179749095/9528128"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,28,41,59,61,66,111,165,182: 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 04:56:00",
  "updated_at": "2026-11-04 04:56:00"
}