{
  "id": 6442,
  "curve_key": "298976990798708058816:-4670862509777057350342725283200",
  "ainvs": [
    "0",
    "0",
    "0",
    "-6228687308306417892",
    "5406090867797520081415191300"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 141.44069487141581,
  "faltings_height": 9.758750931868558,
  "conductor": "68703648682458326225449551581699878684569624",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "43",
    "47",
    "801833",
    "7959004901",
    "137261749107944340721"
  ],
  "discriminant": "2840145690292606695657045392520724522361316133781151282432",
  "regulator": "11560529572165.5972492292136214106607116458769008061633749607987710443595",
  "points": [
    [
      "45023090784",
      "-9538884622518174"
    ],
    [
      "24928965513",
      "-3916927339201701"
    ],
    [
      "-89797833912/49",
      "35423952691552326/343"
    ],
    [
      "-2983328768/9",
      "2328013714282102/27"
    ],
    [
      "869981806",
      "25410783578158"
    ],
    [
      "2730309729/4",
      "-306990601736049/8"
    ],
    [
      "17428023573822/9409",
      "13654991291789063502/912673"
    ],
    [
      "6120745268400/2809",
      "6950459512983439530/148877"
    ],
    [
      "38063147017/16",
      "4073818242752027/64"
    ],
    [
      "18110030464/9",
      "-862383834765046/27"
    ],
    [
      "50815820454/25",
      "-4226912666988858/125"
    ],
    [
      "5262936064488/2401",
      "-5623670778651983574/117649"
    ],
    [
      "5825994898794/2809",
      "-5589059431838376186/148877"
    ],
    [
      "3807786192",
      "-192090006957618"
    ],
    [
      "1142782207974246/573049",
      "-13125692423465636335398/433798093"
    ],
    [
      "-1006074412334/361",
      "229476981278974022/6859"
    ],
    [
      "67215666",
      "-70623857008782"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,2,14,18,33,44,97,106,120,136: 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 17, proved: the 17 listed points come from the point search and are independent, and the 2-descent upper bound (ellrank) is 17. Torsion trivial.",
  "created_at": "2026-11-03 14:18:05",
  "updated_at": "2026-11-03 14:18:05"
}