{
  "id": 175,
  "curve_key": "1398063820612199224758007825:-53934088389538812494987126488734322485625",
  "ainvs": [
    "1",
    "0",
    "0",
    "-29126329596087483849125163",
    "62423713413815642823213203923492112817"
  ],
  "rank_lower_bound": 16,
  "naive_height": 187.5771626848642,
  "faltings_height": 13.547154165453398,
  "conductor": "2199428785729749633481682887731454385883881335809165450",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "17",
    "19",
    "31",
    "43",
    "73",
    "8788437716942464397",
    "4825682552870583764479687"
  ],
  "discriminant": "-101999356220239409114706029907182154313690823961385846723890824782205184000000",
  "regulator": "393519862286889.6501035029965339047136645394249099052522877344608375237",
  "points": [
    [
      "5110339735435422",
      "-365320817187192230444511"
    ],
    [
      "497618622929982",
      "-11099911453301598239391"
    ],
    [
      "138299206337886",
      "-1625188557246978877791"
    ],
    [
      "-1172961026178",
      "9745458965588806689"
    ],
    [
      "1451827418232",
      "4816375036731350409"
    ],
    [
      "1858357791822",
      "3835934257381893489"
    ],
    [
      "2360920940922",
      "2611205774465224089"
    ],
    [
      "2902124502102",
      "1529089488635552049"
    ],
    [
      "30210396731848/9",
      "42492257648703299603/27"
    ],
    [
      "1537958119559502/361",
      "27143958826181392261131/6859"
    ],
    [
      "860963638869174/169",
      "14942816278015024293285/2197"
    ],
    [
      "5129428439172",
      "6926964084183165339"
    ],
    [
      "11766270056735352/529",
      "1241901508300659121973055/12167"
    ],
    [
      "34194892011582",
      "197611265911196429409"
    ],
    [
      "185624555321995496598/5257849",
      "2501114969242211280352058304093/12056247757"
    ],
    [
      "107346062118846/25",
      "508148539017655964541/125"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1907/1: 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 16:01:36",
  "updated_at": "2026-07-01 22:40:38"
}