{
  "id": 194,
  "curve_key": "553199431556088331024:-13049502537014064849315850384832",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-11524988157418506896",
    "15103590903163497035477292096"
  ],
  "rank_lower_bound": 15,
  "naive_height": 143.2926056061889,
  "faltings_height": 9.755185635612015,
  "conductor": "10190033470666423893407916147715865053798375920",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "17",
    "29",
    "31",
    "37",
    "1213",
    "228054274471183",
    "20879094585820301449"
  ],
  "discriminant": "-575294894722450744401258092356189386715188315557054668800",
  "regulator": "14815863059320.462760657204460902116663737179414860667111264912466988",
  "points": [
    [
      "678134618389",
      "-558429614392973290"
    ],
    [
      "773401598709/4",
      "-680050397559446355/8"
    ],
    [
      "12239211117264/529",
      "-42381308250908588520/12167"
    ],
    [
      "12544641506",
      "-1358179764705030"
    ],
    [
      "3569890425248/361",
      "-6390921341055807816/6859"
    ],
    [
      "5591956987984/729",
      "-12102145909255225720/19683"
    ],
    [
      "2898632381921/1681",
      "1303373127022665840/68921"
    ],
    [
      "548698001744/289",
      "39940953579607080/4913"
    ],
    [
      "79509693083361/38416",
      "81170157156022503465/7529536"
    ],
    [
      "62974323862784/29929",
      "67360935645521657880/5177717"
    ],
    [
      "124565294906581/50625",
      "461664469936998565354/11390625"
    ],
    [
      "194522633305405/58564",
      "1644752730648824672287/14172488"
    ],
    [
      "5865379264",
      "386380504805240"
    ],
    [
      "3564819679739280/1907161",
      "25251714986265769716936/2633789341"
    ],
    [
      "157879158605/121",
      "63656031680098026/1331"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2352/1: 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 16:03:13",
  "updated_at": "2026-07-01 22:42:37"
}