{
  "id": 193,
  "curve_key": "4191610887043617416996401:-7297431744382413519811983154967154601",
  "ainvs": [
    "1",
    "0",
    "0",
    "-87325226813408696187425",
    "8446101555990886693844233482455625"
  ],
  "rank_lower_bound": 15,
  "naive_height": 170.08538205546094,
  "faltings_height": 12.167890735676696,
  "conductor": "2282558436678897130581398498347303110857569170",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "19",
    "23",
    "31",
    "41",
    "675304453",
    "3187920097",
    "48610151546889853"
  ],
  "discriminant": "11801171435734791180357853315012481673861744917479863238791703372800000",
  "regulator": "14786943101801.821816404552640176100785119072680650943651152261285874",
  "points": [
    [
      "748436483759858950/1292769",
      "572940742521140768169429475/1469878353"
    ],
    [
      "2457354167530/9",
      "1901167349508952825/27"
    ],
    [
      "1902067649424250/19321",
      "76122157752942148735825/2685619"
    ],
    [
      "101254141450",
      "25340801083372075"
    ],
    [
      "554406704154395050/5193841",
      "218571678161936581007716925/11836763639"
    ],
    [
      "90352672879870/841",
      "425494653096663013775/24389"
    ],
    [
      "109936667050",
      "13212042066356875"
    ],
    [
      "606701606153755/5476",
      "4639386552942000078275/405224"
    ],
    [
      "112298274550",
      "7470676593563575"
    ],
    [
      "144223305244433530/1274641",
      "5382887165331282577854355/1439069689"
    ],
    [
      "13557424351450/121",
      "-11006174098335477775/1331"
    ],
    [
      "77564540933783050/734449",
      "-12614392494911029526074525/629422793"
    ],
    [
      "1126143904425925/4624",
      "12671172546060460859525/314432"
    ],
    [
      "347065469002",
      "141223434230660971"
    ],
    [
      "1471166114339530",
      "-56427728311403536554005"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1265/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 15 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.",
  "created_at": "2026-06-25 16:03:08",
  "updated_at": "2026-07-01 22:42:39"
}