{
  "id": 6177,
  "curve_key": "264712732982647248481:-4292847828330191575812680976145",
  "ainvs": [
    "1",
    "0",
    "0",
    "-5514848603805151010",
    "4968573874922595458725358916"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 141.0755306562504,
  "faltings_height": 9.574770031790784,
  "conductor": "4417713314635932233592727617217483889651187474",
  "bad_primes": [
    "2",
    "3",
    "7",
    "806269",
    "7515550179465343",
    "2479758901030383026513"
  ],
  "discriminant": "69807942982483536339541642883807560246889870614304489472",
  "regulator": "24203061893995598.3223778061784784316470104002763579021949427795376985373",
  "points": [
    [
      "-2662583708",
      "27863972591374"
    ],
    [
      "-1515032732",
      "99229073840398"
    ],
    [
      "37004818",
      "69025691047234"
    ],
    [
      "4962357955/4",
      "48164838623963/8"
    ],
    [
      "1509710500",
      "9149116466446"
    ],
    [
      "6107781871/4",
      "83082097083725/8"
    ],
    [
      "1542800860",
      "11509594820974"
    ],
    [
      "1642771996",
      "18500084937838"
    ],
    [
      "1948476724",
      "40254851857150"
    ],
    [
      "2185489492",
      "57918025282654"
    ],
    [
      "2410730692",
      "75391206339214"
    ],
    [
      "2744328292",
      "102480246469774"
    ],
    [
      "11051636740",
      "1137470950601326"
    ],
    [
      "80605304308",
      "22875064973868238"
    ],
    [
      "98423988154/25",
      "26302776008340932/125"
    ],
    [
      "1438833267124/5329",
      "23012019526399542046/389017"
    ],
    [
      "6317756719684/3025",
      "8418812077492149538/166375"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,28,55,64,75,80,81,86,88,129: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-30 14:40:26",
  "updated_at": "2026-10-30 14:40:26"
}