{
  "id": 6496,
  "curve_key": "15077497053592691257100656:-167602483439904523992599990087472801304",
  "ainvs": [
    "0",
    "1",
    "1",
    "-314114521949847734522930",
    "193984355833118123854488928171215506"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 176.02931670676875,
  "faltings_height": 12.722205255661546,
  "conductor": "2201233101115632161881877774585510484195773539979034855",
  "bad_primes": [
    "3",
    "5",
    "7",
    "47",
    "53",
    "242403607",
    "6898139839909",
    "5033058012181595839842686447"
  ],
  "discriminant": "-14272575936295091134683531097482395186459943673377980502198243939029296875",
  "regulator": "14687330929661136.70455527840581323969178186767021282923860342995440916169851",
  "points": [
    [
      "10125847323628",
      "32175217564641337126"
    ],
    [
      "252395464575685/4",
      "-4009639754307699146779/8"
    ],
    [
      "1149565826424115/121",
      "38912850385741721587397/1331"
    ],
    [
      "3175323084954535/1089",
      "-176304920923406263183156/35937"
    ],
    [
      "1412588029384540/841",
      "-51188287505197298998882/24389"
    ],
    [
      "266442626685265/576",
      "-5312127776444455358087/13824"
    ],
    [
      "364699793413873/841",
      "-9103258389122922585586/24389"
    ],
    [
      "1356917825225365/25921",
      "-1759147968122014443883078/4173281"
    ],
    [
      "-171028276297710/841",
      "-12181157818974462331507/24389"
    ],
    [
      "398126462556881755/97969",
      "249172671705361758386041279/30664297"
    ],
    [
      "53014546420101460/6889",
      "12176729091365002397518019/571787"
    ],
    [
      "170545268101781185/157609",
      "66250778989465612054264126/62570773"
    ],
    [
      "7341481179054940/1681",
      "624572978530094656121777/68921"
    ],
    [
      "249342195018396910/4489",
      "124500703143321859679865931/300763"
    ],
    [
      "23152984955751310/24649",
      "3301187428950805984819366/3869893"
    ],
    [
      "228668966065",
      "366214551688905937"
    ],
    [
      "290736907494105145/222784",
      "148955229794900111822875021/105154048"
    ],
    [
      "967308028781231740/1787569",
      "1020894660841284044741552801/2389979753"
    ],
    [
      "-160861998878712335/929296",
      "441760691726189034578707153/895841344"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,20,22,99,116,144,145,161,174,195: 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 >= 19: the 19 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 04:53:46",
  "updated_at": "2026-11-04 04:53:46"
}