{
  "id": 178,
  "curve_key": "11132809082008253805748920129:-1174891147887394634701529646295101307118817",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-231933522541838620953102503",
    "1359827717462320292062961439163457381087"
  ],
  "rank_lower_bound": 16,
  "naive_height": 193.7394988179177,
  "faltings_height": 13.864110287723491,
  "conductor": "29000334883318273334379766369449958734968994052897662690",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "19",
    "23",
    "3294355577207735591",
    "1011771035177430581876701163"
  ],
  "discriminant": "-332816996738660684218455986579305397186353893376619101695289646827493750937600",
  "regulator": "1597599608068227.671771351581259324452130616281157992166321160531013830",
  "points": [
    [
      "39311985292419612185763/143592289",
      "7782675174704441356447161385367794/1720666399087"
    ],
    [
      "105538367117583",
      "1073501176971972464158"
    ],
    [
      "91326967639593",
      "861336775954602458668"
    ],
    [
      "6271796033530083/289",
      "397534517028079176885710/4913"
    ],
    [
      "990291502852785/64",
      "19668508858275895940171/512"
    ],
    [
      "115917997431619887/10609",
      "12463248179136640291539886/1092727"
    ],
    [
      "1489574132659398015195/209989081",
      "-25735861290825681984151049464694/3042951772771"
    ],
    [
      "1805829793061414733123/213773641",
      "-5752256085260123832006215471362/3125584405061"
    ],
    [
      "2185511165043919158579/247275625",
      "-2262155248862281805639449793842/3888409203125"
    ],
    [
      "2476891705164386445285/274200481",
      "-6129616608644242348739175367258/4540485764879"
    ],
    [
      "56792205463555933683/5958481",
      "-56477693860551678057262392322/14544652121"
    ],
    [
      "187821875570497/9",
      "-2022024040126227255884/27"
    ],
    [
      "63655414911651003319473/2863213081",
      "-12992980561964414368856652135329918/153207668751229"
    ],
    [
      "37679089417444755/961",
      "-6828593700442511025350654/29791"
    ],
    [
      "1147562992021579/25",
      "-36961131093176466599818/125"
    ],
    [
      "3504789305602515987/3721",
      "-6560492695982148303386152730/226981"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=1851/11: 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 16:01:51",
  "updated_at": "2026-07-01 22:40:38"
}