{
  "id": 199,
  "curve_key": "999369977481478619918408638878049:-31737257213015815716596901650851489133526221191057",
  "ainvs": [
    "1",
    "0",
    "0",
    "-20820207864197471248300179976626",
    "36732936589138673862895758597955508398047757956"
  ],
  "rank_lower_bound": 13,
  "naive_height": 227.96315152173966,
  "faltings_height": 16.82881768633617,
  "conductor": "21785392458764315483988614758901932764423833726404496768609056369259382575196330",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "19",
    "86044359449701746413144681",
    "4881201479084572277272794646126863424004819546339"
  ],
  "discriminant": "-5290724783356970045570091261064369041862691109828850503901233859803837652758778996437693235200",
  "regulator": "3144017645707793.395706961347052974878979749568459122697271441584",
  "points": [
    [
      "1162634353995497656800804/7778521",
      "1253039186201942747498661935105222982/21694295069"
    ],
    [
      "3627637297190869531/36",
      "6902364641611559233412383477/216"
    ],
    [
      "1558179052657862437/144",
      "1794466997628157186816731475/1728"
    ],
    [
      "3881968959990696",
      "120039898039176624614682"
    ],
    [
      "2837185351207296",
      "22369929495303810169122"
    ],
    [
      "133369750392561176/49",
      "5181060484570817333704274/343"
    ],
    [
      "290049012608047521903204/131859289",
      "60156526711202358713019503533166394/1514140215587"
    ],
    [
      "27074001988877942159/6724",
      "-74342839994965136661541523599/551368"
    ],
    [
      "2480181137075511944/289",
      "-3439495665410956315384662434/4913"
    ],
    [
      "9363687175740228",
      "-814109553204195618921090"
    ],
    [
      "4354296816133898977943076/338376025",
      "-8579048965788332093253582643648116486/6224426979875"
    ],
    [
      "11397032844750127922575524/4282000969",
      "3685010112080220968145792764888880246/280201297408453"
    ],
    [
      "19369629507101316",
      "-2626899353324391553849218"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2789/9: 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 13 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 16:03:40",
  "updated_at": "2026-07-01 22:46:36"
}