{
  "id": 6499,
  "curve_key": "12527086100883806006521:-1417347228349503199904491380049165",
  "ainvs": [
    "1",
    "0",
    "1",
    "-260980960435079291803",
    "1640448180938324734668015822962"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 152.66819008840056,
  "faltings_height": 10.59057938915655,
  "conductor": "51905558242207374520110452789771714456654440002866497134",
  "bad_primes": [
    "2",
    "3",
    "7",
    "47",
    "3917",
    "6712945886802542643348784749325054461723937871273"
  ],
  "discriminant": "-24898161988738517824555622215103275533996915089455007078231388",
  "regulator": "288039943129525366.0274387334485100741773686063343377704851087876441787278706",
  "points": [
    [
      "1283440331821",
      "1453881780085306430"
    ],
    [
      "335695883611",
      "-194278779231382870"
    ],
    [
      "10163500756795/169",
      "31337087862250437086/2197"
    ],
    [
      "338692734091/25",
      "-96121406460100754/125"
    ],
    [
      "34125359739241/3025",
      "-60440246327042085076/166375"
    ],
    [
      "2730761019490/289",
      "660584989623547249/4913"
    ],
    [
      "58390010243509/7921",
      "241322019804283848838/704969"
    ],
    [
      "97400220469/25",
      "-103289969988603932/125"
    ],
    [
      "76001536181641/23104",
      "-3175304010206754139481/3511808"
    ],
    [
      "16485858648077083/25877569",
      "159845026707635488401886198/131639193503"
    ],
    [
      "113694588715/1156",
      "49945179393266899379/39304"
    ],
    [
      "-148061720",
      "1295795669400038"
    ],
    [
      "3634730844215200/1261129",
      "1352651341093963569044282/1416247867"
    ],
    [
      "-2402231432",
      "1501174826065046"
    ],
    [
      "276431742354373/38416",
      "2767345126035686543495/7529536"
    ],
    [
      "-1772884233132707/121801",
      "65238963399534245664404/42508549"
    ],
    [
      "1074448587688/121",
      "202597759322984131/1331"
    ],
    [
      "9077561932",
      "139235518863506"
    ],
    [
      "11152392913",
      "342009792923306"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,13,45,55,68,89,135,152,180: 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 >= 19: the 19 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 04:54:07",
  "updated_at": "2026-11-04 04:54:07"
}