{
  "id": 170,
  "curve_key": "48116229456189285946801:-10783786994471240316075542370260201",
  "ainvs": [
    "1",
    "1",
    "1",
    "-1002421447003943457225",
    "12481234946887000688835790191735"
  ],
  "rank_lower_bound": 16,
  "naive_height": 156.72670374156405,
  "faltings_height": 10.959174293845557,
  "conductor": "5859468212246645188748237684777485444087182008624730",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "717496229",
    "2773745671",
    "686301029029469989491869608943"
  ],
  "discriminant": "-2831445042543812469319980404169211270210294978426890329088000000",
  "regulator": "755715242422743.0255065980163703580980962319872380216750682669847423293",
  "points": [
    [
      "751898661471/25",
      "-386048269516272384/125"
    ],
    [
      "316496094584587/3249",
      "5364912463885590522304/185193"
    ],
    [
      "21289729305823/81",
      "97551108696840416632/729"
    ],
    [
      "373302054943",
      "227287226501355848"
    ],
    [
      "7617130217783727/12769",
      "663876097852519025472376/1442897"
    ],
    [
      "111751965933",
      "-36001073991025792"
    ],
    [
      "54441827983",
      "-10921034534816392"
    ],
    [
      "49313028731983/961",
      "-292136740323616231672/29791"
    ],
    [
      "5650946185995727/127449",
      "-338053143254526723960056/45499293"
    ],
    [
      "9552398586646567/499849",
      "-194798114168374305739256/353393243"
    ],
    [
      "41878570208387/2209",
      "56001451768964000304/103823"
    ],
    [
      "25153429663",
      "1783620616336328"
    ],
    [
      "1427445371977/49",
      "970238560834856594/343"
    ],
    [
      "272873933446253137/7360369",
      "102353737160916667605018296/19968681097"
    ],
    [
      "33618451033",
      "-4095985604896642"
    ],
    [
      "8651547690714212467/123587689",
      "23208991631678744562007616404/1373924338613"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=771/5: 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 16:01:11",
  "updated_at": "2026-07-01 22:37:40"
}