{
  "id": 192,
  "curve_key": "40874711982984207510769:-8196790483963389560660650305834697",
  "ainvs": [
    "1",
    "0",
    "0",
    "-851556499645504323141",
    "9487026023034811912831219901121"
  ],
  "rank_lower_bound": 15,
  "naive_height": 156.19439560733912,
  "faltings_height": 10.872367536621327,
  "conductor": "562577168211394579270024943629595507978443710",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "19",
    "103",
    "109",
    "6211",
    "64376373383029",
    "19987699093727394221"
  ],
  "discriminant": "638730941704979414939006251941574649893567559880993486621081600",
  "regulator": "6081709767502.5731747914878712722800396843622123174768992444677519175",
  "points": [
    [
      "84305679874/9",
      "-41233185475744451/27"
    ],
    [
      "36241602249954/13225",
      "-4073566388142236145363/1520875"
    ],
    [
      "-14513645610267290/2047761",
      "-11411968604968034879033251/2930345991"
    ],
    [
      "5854928580399234/1075369",
      "-2496576161553282539156589/1115157653"
    ],
    [
      "2941002666",
      "-2647270682374113"
    ],
    [
      "600692104935810/326041",
      "-524072430071294943344139/186169411"
    ],
    [
      "-7819076214",
      "-3958198824213513"
    ],
    [
      "-23145631818534/1681",
      "-297253625339500924833/68921"
    ],
    [
      "7156924540410/49",
      "18790187467681606353/343"
    ],
    [
      "2536220959236",
      "-4038798111452968923"
    ],
    [
      "124435835730",
      "-42782323227585561"
    ],
    [
      "41205269280306/961",
      "-214420931827589747943/29791"
    ],
    [
      "21031135188",
      "-938135835012105"
    ],
    [
      "42768585876114/2809",
      "-33673699761100504821/148877"
    ],
    [
      "5156498703514/441",
      "-9838663942889014997/9261"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-125, -99, -26, -18, 125, 143] with shift t=593/6: 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 15 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 16:03:03",
  "updated_at": "2026-07-01 22:42:38"
}