{
  "id": 187,
  "curve_key": "57086385615639196900478739409:-13643169732991798117322623791734476775741977",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1189299700325816602093307071",
    "15790705709481148712666898533494358863001"
  ],
  "rank_lower_bound": 16,
  "naive_height": 198.6436258325694,
  "faltings_height": 14.28113104372204,
  "conductor": "6478350825852857868575711796470819320010194843359558272176170",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "23",
    "31",
    "43",
    "3277612446733284530993",
    "1816531268684434524436436201159"
  ],
  "discriminant": "-57756388763308488291464979237750795586612877872758868116877248909627520480051200",
  "regulator": "6104323411601247.835938464639324428147234046756706737038186530028901018",
  "points": [
    [
      "697576442233977388406/177241",
      "-18423455744863744566579326547943/74618461"
    ],
    [
      "1617980147791814",
      "-65067171076314760848307"
    ],
    [
      "8996486721959734/9",
      "-852813860149243222627361/27"
    ],
    [
      "2089299406246950440/18769",
      "-2889502216708937627799054803/2571353"
    ],
    [
      "4959252092193526/361",
      "310190738443800799859023/6859"
    ],
    [
      "810579905488502486/52441",
      "398415122436772395438033593/12008989"
    ],
    [
      "27510586103866689230/1530169",
      "28071482427182809558959004193/1892819053"
    ],
    [
      "18427107540326",
      "11507788246080029357"
    ],
    [
      "3346688353809670/169",
      "4876157311998696125137/2197"
    ],
    [
      "184998133634752/9",
      "146224271703375470585/27"
    ],
    [
      "23177380051814",
      "26010241714416410093"
    ],
    [
      "39043352706710",
      "169922010875689734941"
    ],
    [
      "59337157252816",
      "392608088389820811787"
    ],
    [
      "30436825206632108486/398161",
      "153149547828250401696053592347/251239591"
    ],
    [
      "17545171071416",
      "18033907841991859187"
    ],
    [
      "19600296319046",
      "3155410596522222317"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=842/9: 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 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 16:02:37",
  "updated_at": "2026-07-01 22:40:40"
}