{
  "id": 6074,
  "curve_key": "30743334938754205264:-169432346413942576106447332288",
  "ainvs": [
    "0",
    "1",
    "0",
    "-640486144557379276",
    "196102252580419637122595024"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 134.6166146840774,
  "faltings_height": 9.061723530986006,
  "conductor": "1558190906638665417508116153288187194306360",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "23971",
    "118730017682438237",
    "350953342328176603"
  ],
  "discriminant": "202444598074402612402085312538318584306422115912121600",
  "regulator": "507942562301564194.0032458377865199101955425959373781386982239597971951",
  "points": [
    [
      "-823844858",
      "12829777849278"
    ],
    [
      "432399404",
      "42608837700"
    ],
    [
      "3390797747",
      "192380020079328"
    ],
    [
      "-2680347262748462/3523129",
      "103092694677618722320374/6612913133"
    ],
    [
      "-277345190078704/3087049",
      "86259375957619826757900/5423945093"
    ],
    [
      "10637862526/25",
      "97697590753074/125"
    ],
    [
      "49791215246/121",
      "1984367424091650/1331"
    ],
    [
      "108515182934/361",
      "38026268723517750/6859"
    ],
    [
      "124952444615/289",
      "341857268934246/4913"
    ],
    [
      "260555510690/529",
      "4311564204985986/12167"
    ],
    [
      "549495267494/961",
      "122192776698343650/29791"
    ],
    [
      "823368292397/1156",
      "395481110506524837/39304"
    ],
    [
      "1476270180506/1225",
      "1469335056120816354/42875"
    ],
    [
      "6640215145046/18769",
      "9548206410151514430/2571353"
    ],
    [
      "383606550908912/73441",
      "7429787144464736244492/19902511"
    ],
    [
      "4441687626537206/477481",
      "294959298348318771438570/329939371"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,11,30,37,51,101,102,104,129: 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": "2026-10-28 21:15:01",
  "updated_at": "2026-10-28 21:15:01"
}