{
  "id": 6173,
  "curve_key": "24187085806383533761:-130651848362149973891789089825",
  "ainvs": [
    "1",
    "0",
    "0",
    "-503897620966323620",
    "151217417043830260627562256"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 134.084667300278,
  "faltings_height": 9.14083469858599,
  "conductor": "2206369951742974882400755286879288413262834238",
  "bad_primes": [
    "2",
    "3",
    "19",
    "109089012578454890341",
    "177415872838624908605387"
  ],
  "discriminant": "-1689869548427228196035762708318731556714076077720629248",
  "regulator": "49108554686216799909.3828171496952221845606290244191053930535068278937549",
  "points": [
    [
      "2729912911/4",
      "89511730498685/8"
    ],
    [
      "4115379736",
      "260338105334428"
    ],
    [
      "-987992467973296856/1298232961",
      "453382296848172943374313060/46776631817791"
    ],
    [
      "-6517336747208/7921",
      "2091669941071973756/704969"
    ],
    [
      "38986164244/9",
      "7600920370335812/27"
    ],
    [
      "134280730696/289",
      "20491190196003740/4913"
    ],
    [
      "3125322374536/4489",
      "3531334755784887988/300763"
    ],
    [
      "5912591814184/16129",
      "8131429671468659908/2048383"
    ],
    [
      "6655357818136/14641",
      "7105793877283411708/1771561"
    ],
    [
      "78705008295652/80089",
      "557513846236397041828/22665187"
    ],
    [
      "149067441950500/426409",
      "1174205178846771668732/278445077"
    ],
    [
      "171623317705480/22201",
      "2239210210041517946924/3307949"
    ],
    [
      "245094630455604/112225",
      "3658026373062048007872/37595375"
    ],
    [
      "1237548612228514/2486929",
      "19088446110347041731974/3921887033"
    ],
    [
      "3175527649113052/3038049",
      "146603015432508595496488/5295319407"
    ],
    [
      "10631995715997316/990025",
      "1093945966761634175085104/985074875"
    ],
    [
      "459938989482922888/103286569",
      "308204464462005487822534388/1049701400747"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,19,29,40,50,101,105,118,131: 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 14:39:56",
  "updated_at": "2026-10-30 14:39:56"
}