{
  "id": 6508,
  "curve_key": "5498231298139837662321:-406386805931917856712178084694025",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-114546485377913284632",
    "470355099486874881428017771035"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 150.17614023371507,
  "faltings_height": 10.332536132982513,
  "conductor": "58355173931578483333998489018706959609875067416042",
  "bad_primes": [
    "2",
    "11",
    "274962639936119",
    "88016446198598609",
    "109602174072609641"
  ],
  "discriminant": "615917755882186101189757544362919497048390990719941354586112",
  "regulator": "1933466697304407.6050319846549472013884767410311885133109484085654778908258",
  "points": [
    [
      "43917729901",
      "-8952496567877303"
    ],
    [
      "40348504813",
      "-7844509691391671"
    ],
    [
      "-12211713557",
      "-219266799220173"
    ],
    [
      "443069501395/81",
      "62934396369837011/729"
    ],
    [
      "472187536957/81",
      "-19440281779475615/729"
    ],
    [
      "6758134900301/1225",
      "-3410842457819398181/42875"
    ],
    [
      "101167570432901/15625",
      "22427942517021617149/1953125"
    ],
    [
      "1346738780633/169",
      "554032673800003445/2197"
    ],
    [
      "232488208016333/28561",
      "1342013571702192291553/4826809"
    ],
    [
      "6484774957",
      "-15708721449015"
    ],
    [
      "8203317100609/1225",
      "2568125397970693707/42875"
    ],
    [
      "6372922570634203/956484",
      "-50698551203457604414571/935441352"
    ],
    [
      "-70194142691/9",
      "25461977314215019/27"
    ],
    [
      "-223389226764571/24025",
      "-3185020083793529988293/3723875"
    ],
    [
      "-544283997171/2209",
      "-73308413498812092089/103823"
    ],
    [
      "2488479684553908037/382710969",
      "-159799606131125445021966037/7486974686547"
    ],
    [
      "6965888269",
      "-102214306077335"
    ],
    [
      "2514595437333/529",
      "-2219226779831861625/12167"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,4,12,33,35,125,143,169,184: 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 >= 18: the 18 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.",
  "created_at": "2026-11-04 04:55:12",
  "updated_at": "2026-11-04 04:55:12"
}