{
  "id": 5972,
  "curve_key": "409354623647403706638121:-266253676418262502677661627305897685",
  "ainvs": [
    "1",
    "0",
    "1",
    "-8528221325987577221628",
    "308163977335241285581239029798542"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 163.13951518775104,
  "faltings_height": 11.481585642389877,
  "conductor": "1031701942448500238814623346383994187948218724925406",
  "bad_primes": [
    "2",
    "3",
    "7",
    "17",
    "41",
    "47",
    "83",
    "419",
    "1247547913",
    "31236385927",
    "79043498212870820693"
  ],
  "discriminant": "-1328108484709708764957458296805164523651599807998227862760891234788",
  "regulator": "890518180517049.33818061773355008335483680147052148956871052141577687607672",
  "points": [
    [
      "-103497883303",
      "9064652142135801"
    ],
    [
      "-96874552981",
      "15006547747287153"
    ],
    [
      "-18614461330",
      "21458395700991558"
    ],
    [
      "8951791328",
      "15249207554787690"
    ],
    [
      "9100298330",
      "15208817368758903"
    ],
    [
      "14742622976",
      "13624964557530453"
    ],
    [
      "43791891497",
      "4321788832778481"
    ],
    [
      "43999330274",
      "4255334725032063"
    ],
    [
      "45646093385",
      "3740382402074613"
    ],
    [
      "46836927638",
      "3387358592189811"
    ],
    [
      "48717515561",
      "2883756917638245"
    ],
    [
      "60778716191",
      "3788028827243613"
    ],
    [
      "65308442147",
      "5454526304797011"
    ],
    [
      "93860695961",
      "18291918152311245"
    ],
    [
      "102137457860",
      "22419075812676201"
    ],
    [
      "120514957235",
      "32104884894306201"
    ],
    [
      "254696359103",
      "121071214476932931"
    ],
    [
      "1145206425119/121",
      "20110707554018285583/1331"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "The curve is the fibre at T = 4/13 from curve #4220:\r\n\r\n- A4(4/13) = −250530051225379351163799381830307 / 13⁸\r\n- A6(4/13) = 1551614814321580263454294680252020288318239844190 / 13¹²\r\n\r\nIntegral model. The denominators are 13⁸ and 13¹². Scaling x and y by 13⁴ and 13⁶ clears them, which gives\r\n\r\ny² = x³ − 250530051225379351163799381830307 x + 1551614814321580263454294680252020288318239844190.\r\n\r\nIts minimal model is the record curve:\r\n\r\n[1, 0, 1, −8528221325987577221628, 308163977335241285581239029798542]\r\n\r\nRank 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-27 11:29:23",
  "updated_at": "2026-10-27 11:29:23"
}