{
  "id": 203,
  "curve_key": "331074711495990018937:-6146034258883033837093628371789",
  "ainvs": [
    "1",
    "0",
    "1",
    "-6897389822833125395",
    "7113465576836136307696346462"
  ],
  "rank_lower_bound": 13,
  "naive_height": 141.7867196553794,
  "faltings_height": 9.711030945984817,
  "conductor": "2840704603917455365782151764722058186",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "17",
    "19",
    "23",
    "37",
    "131",
    "177023219",
    "816206070761378053"
  ],
  "discriminant": "-859076487022818729321026088704523026210935982019962548656",
  "regulator": "2084640105.287431943944479547655984090376395520069190176449159295",
  "points": [
    [
      "1821696636986571/1681",
      "-77752259635965980361424/68921"
    ],
    [
      "2651704780659/100",
      "-4297661292794605493/1000"
    ],
    [
      "11101674264",
      "-1139650601752094"
    ],
    [
      "6952530720",
      "-543352700813627"
    ],
    [
      "10172211834471/2209",
      "-28051427564651955172/103823"
    ],
    [
      "2670689975",
      "-87987691259172"
    ],
    [
      "15395621611/9",
      "-483250003213552/27"
    ],
    [
      "26665787397/16",
      "-1006638459099257/64"
    ],
    [
      "1553029431",
      "-12140319849908"
    ],
    [
      "248019741134157063/172318129",
      "-29292387555156537782663404/2262020079383"
    ],
    [
      "208894998531/9409",
      "-76143106562997552752/912673"
    ],
    [
      "7618219107",
      "629843965912912"
    ],
    [
      "71190258388521651/5313025",
      "18654730438432478327441884/12246522625"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=296/9: 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 16:04:00",
  "updated_at": "2026-07-01 22:45:38"
}