{
  "id": 6109,
  "curve_key": "4445226533375405121:-9013510893765111572587685985",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-92608886111987607",
    "10432304298343322885158575"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 128.8150875155973,
  "faltings_height": 8.6611420520371,
  "conductor": "201612979168923207004963766111743393408646",
  "bad_primes": [
    "2",
    "7",
    "11",
    "31",
    "42231457722857814621902757878454837329"
  ],
  "discriminant": "3816244637494318406672231279804842579593723788787712",
  "regulator": "40727474448249963.23102231356193332333669976788330610868288011668481083",
  "points": [
    [
      "-62983737",
      "4001944111562"
    ],
    [
      "258895381",
      "1951594163872"
    ],
    [
      "-46587839153837/1739761",
      "8239733690641210664466/2294744759"
    ],
    [
      "-4914648172965/17689",
      "9025190683221763590/2352637"
    ],
    [
      "-1864106988677/5329",
      "60555895170184654/389017"
    ],
    [
      "-308072087005/9409",
      "3344613276463858270/912673"
    ],
    [
      "-14231438093/49",
      "1228633858187826/343"
    ],
    [
      "927064639/9",
      "38047343304602/27"
    ],
    [
      "4393830547/49",
      "578941298516946/343"
    ],
    [
      "37190323323/169",
      "1850415261704534/2197"
    ],
    [
      "201534159859/289",
      "82935019973891598/4913"
    ],
    [
      "237389119819/121",
      "114341769512822810/1331"
    ],
    [
      "3934253068171/1681",
      "7740444687262409486/68921"
    ],
    [
      "12224340193171/57121",
      "8789955934756082834/13651919"
    ],
    [
      "12316457697707/85849",
      "7910625327972542918/25153757"
    ],
    [
      "444554121697171/2187441",
      "235215936812979430994/3235225239"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,10,54,60,61,66,67,74,81,103: 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 02:29:54",
  "updated_at": "2026-10-29 02:29:54"
}