{
  "id": 168,
  "curve_key": "839985266498485472928414289:-21120087264175663299662184507972168126937",
  "ainvs": [
    "1",
    "0",
    "0",
    "-17499693052051780686008631",
    "24444545444646300140928746272873582761"
  ],
  "rank_lower_bound": 16,
  "naive_height": 185.98627975097364,
  "faltings_height": 13.486825864689092,
  "conductor": "100907360654491808490244547862425873324272623451710",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "31",
    "157",
    "821431020607251773",
    "2908286759476644848543"
  ],
  "discriminant": "84846485315177942827767106512600101143726929143544805106516597149399675699200",
  "regulator": "512847700234413.5846609390960665788755182046399204228090518568879266448",
  "points": [
    [
      "8749030777802664/2809",
      "-58499245138940147954151/148877"
    ],
    [
      "9742749700101321/3136",
      "-44196140114557026187593/175616"
    ],
    [
      "3123962677398",
      "-513183191869000203"
    ],
    [
      "5367228943899126/1681",
      "-72934916594247767556723/68921"
    ],
    [
      "5547244131698613186/1692601",
      "-3335257317853511364588473523/2202073901"
    ],
    [
      "329295175913651664/94249",
      "-70598302430522311861725759/28934443"
    ],
    [
      "946750469202830567910/247464361",
      "-14298856189125828821136719305689/3892861862891"
    ],
    [
      "22196650584696645/5776",
      "-1639371507717248414470239/438976"
    ],
    [
      "312234688643535594/69169",
      "-111299518938285523319516481/18191447"
    ],
    [
      "1414957717465494/289",
      "-36808103808527122498779/4913"
    ],
    [
      "9251504323176",
      "-25580962158180935973"
    ],
    [
      "45938210681459/4",
      "-292653878256911864789/8"
    ],
    [
      "2429484162389272848/11881",
      "-3786002067932057242550465007/1295029"
    ],
    [
      "4031610336891147702/5329",
      "8094894445441753966286486445/389017"
    ],
    [
      "138398804253815111544/6436369",
      "1599095523484338264148228166931/16329068153"
    ],
    [
      "621178546558602/49",
      "14712769473687372255771/343"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=13506/11: 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 16:01:01",
  "updated_at": "2026-07-01 22:39:41"
}