{
  "id": 204,
  "curve_key": "1959173408006340874297187670049:-2743405664101790044589728974331337298125159857",
  "ainvs": [
    "1",
    "0",
    "0",
    "-40816112666798768214524743126",
    "3175238037154846187302797412986141032805156"
  ],
  "rank_lower_bound": 12,
  "naive_height": 209.25105855284116,
  "faltings_height": 15.179948694345056,
  "conductor": "8207915399785028827911225140199659880348230953515903817573863786069230",
  "bad_primes": [
    "2",
    "5",
    "7",
    "17",
    "31",
    "157",
    "1417178528658108860031670181482848799986917717765832284580576751"
  ],
  "discriminant": "-3623222374799711246641358701859136275557470213922538013251076487946923600073999257600",
  "regulator": "51826916814533.727636603725739418132382839118508075688787171474",
  "points": [
    [
      "9770649366740959079884/619369",
      "-965716610723945297620503113535234/487443403"
    ],
    [
      "60233506353420124/9",
      "-14776159007078725940508866/27"
    ],
    [
      "1632143825512484940100/3374569",
      "-60920056763901273997043006329398/6199083253"
    ],
    [
      "28290587106843356/361",
      "4641255380757530029904158/6859"
    ],
    [
      "110391515971620",
      "121441937340674347746"
    ],
    [
      "19680454388442956/169",
      "80229753769020194626786/2197"
    ],
    [
      "17387195299617109/144",
      "147427941541656072822491/1728"
    ],
    [
      "150037170137836",
      "654838533101906691642"
    ],
    [
      "244244363437608",
      "2788656466736583932670"
    ],
    [
      "5484612139645565/16",
      "347390158672825672194889/64"
    ],
    [
      "22903392289503081044364/49042009",
      "3184412100432593479561128806746734/343441189027"
    ],
    [
      "419339596095967/4",
      "1761051162359423345641/8"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=787/7: 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 16:04:04",
  "updated_at": "2026-07-01 22:47:35"
}