{
  "id": 180,
  "curve_key": "9446505198390597638003569:-27870123960027752077586111949178242697",
  "ainvs": [
    "1",
    "0",
    "0",
    "-196802191633137450791741",
    "32257087916682386944199682931390321"
  ],
  "rank_lower_bound": 16,
  "naive_height": 172.52306125405872,
  "faltings_height": 12.305699569019374,
  "conductor": "118450782829775208067766915296945483495715377841216329370",
  "bad_primes": [
    "2",
    "5",
    "11",
    "13",
    "17",
    "19",
    "23",
    "773",
    "429409",
    "1975931940579152131201026051304307748559"
  ],
  "discriminant": "38326897051202141276118593252174781295683024000118658651028382352998400",
  "regulator": "2729177115059906.931362570259210191760555095033069027438437708347161738",
  "points": [
    [
      "960455977590754/9",
      "29765448306343424415989/27"
    ],
    [
      "1184996775243730/81",
      "40773516619417655092631/729"
    ],
    [
      "93926389235815/36",
      "897876538745062245611/216"
    ],
    [
      "1581689387006386/841",
      "61286211837482843133963/24389"
    ],
    [
      "969908831278",
      "868211808480707547"
    ],
    [
      "4642405601764/9",
      "7040144543520120709/27"
    ],
    [
      "83217564981347554/263169",
      "5474064368994652222878799/135005697"
    ],
    [
      "502451208525362754/1682209",
      "-24075177825653767340300449/2181825073"
    ],
    [
      "36260210765869890/117649",
      "-1195938362197546351593943/40353607"
    ],
    [
      "45170047429741794/134689",
      "-3116501815532706470874079/49430863"
    ],
    [
      "396119718018",
      "-128278541487042673"
    ],
    [
      "412971536906",
      "-146334814869525793"
    ],
    [
      "2289306880424454322/6235009",
      "1517199586293175196745070511/15568817473"
    ],
    [
      "969779882324404774/2712609",
      "-389292386971292995061885239/4467667023"
    ],
    [
      "60899311327155256386/19351201",
      "470748801555341867901936180193/85125933199"
    ],
    [
      "146178188707299250/175561",
      "-49106110071703841753789451/73560059"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2594/1: 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 16 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 16:02:01",
  "updated_at": "2026-07-01 22:39:38"
}