{
  "id": 173,
  "curve_key": "136036556786062058818709425:-1586606540190417935477519966720894875625",
  "ainvs": [
    "1",
    "1",
    "1",
    "-2834094933042959558723113",
    "1836350162256247293173176506110827031"
  ],
  "rank_lower_bound": 16,
  "naive_height": 180.52489764411132,
  "faltings_height": 12.705572400524835,
  "conductor": "29160913962360673855189378606441749566045554275575550",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "23",
    "37",
    "89",
    "907",
    "817719259139",
    "1216105912003",
    "36958698339265441"
  ],
  "discriminant": "95309530954517651197679942445741588992280745806742990246093888224000000",
  "regulator": "344112794184068.5132837095483207356329042306934301174297818860928244136",
  "points": [
    [
      "23133397748770775/6889",
      "-3141471217094483134890456/571787"
    ],
    [
      "3041032595591",
      "-4619620262735245200"
    ],
    [
      "94677769838153919/57121",
      "-17760474203161028032626232/13651919"
    ],
    [
      "1407320880095",
      "-796958327588913048"
    ],
    [
      "2086667697367268975/1560001",
      "-1290511508082994089417956712/1948441249"
    ],
    [
      "1235676276895",
      "-470187049057551048"
    ],
    [
      "317369273758475/289",
      "-1080809551567787511744/4913"
    ],
    [
      "896313884656722515/917764",
      "1685321961566180587914459/879217912"
    ],
    [
      "949674970078745/961",
      "796634291908532022882/29791"
    ],
    [
      "357533678862075/361",
      "210056649647529768208/6859"
    ],
    [
      "1024047828325",
      "89406907895428362"
    ],
    [
      "80539355466799265/76729",
      "2852679029117887766832456/21253933"
    ],
    [
      "67970791480505/64",
      "79871140889193934449/512"
    ],
    [
      "1405423329526175/289",
      "49876486941519954749256/4913"
    ],
    [
      "547498095067505304755/11377129",
      "-12802996373789791304733627991116/38375056117"
    ],
    [
      "209690757267880795/212521",
      "2350255655177088892822812/97972181"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=3199/2: 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 16:01:27",
  "updated_at": "2026-07-01 22:39:40"
}