{
  "id": 163,
  "curve_key": "1086540401236914196090781281:-35911769270128710395765526001926443074321",
  "ainvs": [
    "1",
    "0",
    "0",
    "-22636258359102379085224610",
    "41564547766350787714309803970639244100"
  ],
  "rank_lower_bound": 17,
  "naive_height": 186.7637674067169,
  "faltings_height": 13.374448469288719,
  "conductor": "253719162614603331010718284712814989331353661834356557470",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "53",
    "113",
    "733",
    "161647205569",
    "98509003985351",
    "326103671223493957373"
  ],
  "discriminant": "-4003543597440842456820042201225551911119601184817241350745392322788249600000",
  "regulator": "16709644970985311.2146576317744156672737110289849683945374146824957952763",
  "points": [
    [
      "41963813756980",
      "-270163542135951662210"
    ],
    [
      "36238334888500",
      "-216356341349008698290"
    ],
    [
      "182911406772646514980/24433249",
      "-2062545615877669218157217269870/120773549807"
    ],
    [
      "3761563658380",
      "-3104924371751178170"
    ],
    [
      "98561136705675460/26569",
      "-12731169053892275162899910/4330747"
    ],
    [
      "5138189206478380/1849",
      "-27554611597486805688470/79507"
    ],
    [
      "66765098900596/25",
      "-49867050842311805146/125"
    ],
    [
      "2338487488420",
      "-1190805577977135410"
    ],
    [
      "1512239740180",
      "-3285026256352586210"
    ],
    [
      "479603888121063940/94249",
      "220635347202511675210409050/28934443"
    ],
    [
      "372914677744663396/9025",
      "226279698064107559786550834/857375"
    ],
    [
      "84032515043620",
      "769111086316879443790"
    ],
    [
      "363489079868260",
      "6929470060065718088590"
    ],
    [
      "421895389051189638601540/64821669201",
      "-215147281405435544470104279938143810/16503661800243801"
    ],
    [
      "9783488506214227063300/49042009",
      "-967427023992792791305830551022230/343441189027"
    ],
    [
      "120624113779620",
      "-1323787532231407812210"
    ],
    [
      "466217485868799160/289",
      "-318332482341864215809401190/4913"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-324, 24, 120, 180, 276, -276] with shift t=1355/9: 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 17 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 16:00:14",
  "updated_at": "2026-07-01 22:36:39"
}