{
  "id": 6135,
  "curve_key": "1026055691064501721:-1038326839979121555148941085",
  "ainvs": [
    "1",
    "0",
    "1",
    "-21376160230510453",
    "1201767174120414373324292"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 124.41676109687481,
  "faltings_height": 8.140453532511607,
  "conductor": "110588158636195104405800630734472822633358",
  "bad_primes": [
    "2",
    "3",
    "31",
    "3241316820388679",
    "183431619188777602051957"
  ],
  "discriminant": "1214602353351415357930810588628618740883951116812",
  "regulator": "17570571023746592.41945437728140375101951311670366780645917423710171186",
  "points": [
    [
      "52965829",
      "467039201948"
    ],
    [
      "66046672",
      "279339976331"
    ],
    [
      "71635555",
      "195112499228"
    ],
    [
      "198089965",
      "2177133630686"
    ],
    [
      "236743753",
      "3067456410656"
    ],
    [
      "-51678049283/361",
      "7905457690543754/6859"
    ],
    [
      "-18617130425/24964",
      "4352542570395080947/3944312"
    ],
    [
      "808942726/49",
      "316853228263034/343"
    ],
    [
      "10795370881/49",
      "919427403421736/343"
    ],
    [
      "49091531251/625",
      "1338177868099676/15625"
    ],
    [
      "85729265488/841",
      "6982332518635639/24389"
    ],
    [
      "142210080379/1369",
      "16188370014606164/50653"
    ],
    [
      "226236181009/2209",
      "30573700441588810/103823"
    ],
    [
      "1260654416473/16384",
      "234731894310971677/2097152"
    ],
    [
      "15456987590539/177241",
      "2151148467668352092/74618461"
    ],
    [
      "45012800156353/552049",
      "12283056930464443432/410172407"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,12,13,15,19,32,62,81,102: 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 12:56:33",
  "updated_at": "2026-10-29 12:56:33"
}