{
  "id": 6445,
  "curve_key": "111158312802438270096:-1222942336468020161368212798144",
  "ainvs": [
    "0",
    "0",
    "0",
    "-2315798183384130627",
    "1415442519060208520102098146"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 138.55762499233094,
  "faltings_height": 9.4765461572188,
  "conductor": "636729961016074618212602433984860064248378160",
  "bad_primes": [
    "2",
    "3",
    "5",
    "137",
    "269032733",
    "23993693057739281305146401684243"
  ],
  "discriminant": "-70657923773953800383052492099299921329642524415200000000",
  "regulator": "44808456261264.1459138801005183335127817129610410063331179348932280130778",
  "points": [
    [
      "47005479823/9",
      "-9801729695831950/27"
    ],
    [
      "136035659895",
      "-50170923844711434"
    ],
    [
      "-1110580492473/841",
      "1136310910130814750/24389"
    ],
    [
      "-115044300417/289",
      "234295372062740250/4913"
    ],
    [
      "276332956999/2025",
      "3024976885757888318/91125"
    ],
    [
      "-697491162657/1369",
      "-2513871374091558750/50653"
    ],
    [
      "677219868783/11449",
      "-43805654968315973310/1225043"
    ],
    [
      "766390705917/961",
      "259389358061692500/29791"
    ],
    [
      "1686137745813/1849",
      "625889994996927420/79507"
    ],
    [
      "993244047",
      "-9754922895750"
    ],
    [
      "299941752258/289",
      "-55995979846375350/4913"
    ],
    [
      "733649922",
      "-10551716277750"
    ],
    [
      "21745253469/25",
      "-961848448217172/125"
    ],
    [
      "4905832520517/5041",
      "-3269278240139566260/357911"
    ],
    [
      "1736578222",
      "-51292139560300"
    ],
    [
      "828149806",
      "-8098544578780"
    ],
    [
      "370257695216253/310249",
      "-3239666361915744599100/172808693"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,29,34,45,74,107,124,172,176: 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 14:18:29",
  "updated_at": "2026-11-03 14:18:29"
}