{
  "id": 206,
  "curve_key": "17955918111461417803920655156129:-73176126388097287413385955954358513712822872817",
  "ainvs": [
    "1",
    "0",
    "0",
    "-374081627322112870915013649086",
    "84694590726964421851283320326657184990555716"
  ],
  "rank_lower_bound": 12,
  "naive_height": 215.89641765065846,
  "faltings_height": 15.917906104017392,
  "conductor": "431184740183886106535857306115602438132236071442111719243984779367694673530",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "19",
    "31",
    "157",
    "4517",
    "21447358407501679739",
    "229195053659697773915482188781052624679234107"
  ],
  "discriminant": "251453575815775763875321183856781238190717796098320943436148000167676873962388406215526400",
  "regulator": "128581143130572.87247198242309993116966404006654526604274397426",
  "points": [
    [
      "1311962006756715124/9",
      "1502720013154061565338633534/27"
    ],
    [
      "1622964775848764290/81",
      "2066633657734054159603682216/729"
    ],
    [
      "2072780699018820625/576",
      "2943546468634289028128158229/13824"
    ],
    [
      "2186299717178498316/841",
      "3149957659531808431157396778/24389"
    ],
    [
      "21543146691997293/16",
      "2877874286932028601295563/64"
    ],
    [
      "25660979318382673047964/62520649",
      "-270616637356825393764260426373754/494350771643"
    ],
    [
      "29542255198291148257780/69839449",
      "-854884756696283163043353677345782/583648275293"
    ],
    [
      "36612916576974039331564/79798489",
      "-2214006161797201672783134021818326/712839902237"
    ],
    [
      "539247600876988",
      "-6307035117458519888078"
    ],
    [
      "561618735181336",
      "-7193514911777232202958"
    ],
    [
      "29051022192219566367268/57623281",
      "2153653151864037845479171859549542/437418326071"
    ],
    [
      "3094088113068047834300476/664041361",
      "5397713329122196810957917667676278178/17111681831609"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=7907/3: 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 12 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 16:04:14",
  "updated_at": "2026-07-01 22:47:36"
}