{
  "id": 6134,
  "curve_key": "683038483258943952:-564571801213007304761913024",
  "ainvs": [
    "0",
    "0",
    "0",
    "-14229968401227999",
    "653439584737276973104066"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 123.1962196034252,
  "faltings_height": 7.967303271925952,
  "conductor": "92078813233509391764925470080562121674696",
  "bad_primes": [
    "2",
    "3",
    "61",
    "1163713",
    "549450332688841",
    "32788613704170461"
  ],
  "discriminant": "-43676295953929910852207687776133514546688601856",
  "regulator": "1968292446030769.512723719808331036231541629651249416484095977455984301",
  "points": [
    [
      "-36551878",
      "1060536133494"
    ],
    [
      "36922439",
      "422338216032"
    ],
    [
      "44702867",
      "326574661536"
    ],
    [
      "67671767",
      "19317698136"
    ],
    [
      "82894229",
      "208473357222"
    ],
    [
      "87243203",
      "275703509736"
    ],
    [
      "111000717",
      "664500106814"
    ],
    [
      "112295747",
      "686703443256"
    ],
    [
      "345277745",
      "6074786320794"
    ],
    [
      "966485213",
      "29827639903374"
    ],
    [
      "273364041/4",
      "93097786099/8"
    ],
    [
      "1728035681/25",
      "1187555594454/125"
    ],
    [
      "11683544757/169",
      "21018484774858/2197"
    ],
    [
      "15276695825/16",
      "1874102861205369/64"
    ],
    [
      "30873825149/25",
      "5400402113299518/125"
    ],
    [
      "42773141063/529",
      "2158611317286552/12167"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,21,23,24,30,47,59,61,70,95: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-29 12:56:26",
  "updated_at": "2026-10-29 12:56:26"
}