{
  "id": 213,
  "curve_key": "3577957125609504224117667787414369:-210873622699844738147045972586854887727236925732497",
  "ainvs": [
    "1",
    "0",
    "0",
    "-74540773450198004669118078904466",
    "244066692939635107384386902977544619850388014596"
  ],
  "rank_lower_bound": 12,
  "naive_height": 231.78030021311164,
  "faltings_height": 17.19719209430338,
  "conductor": "10465426762586441668769337684707368604491256699170701827399830238043087368190",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "19",
    "157",
    "6651361",
    "21871169",
    "117189223",
    "2264709407",
    "136116631176129536978806192189908923"
  ],
  "discriminant": "773452069750637843410919255151077621482402538193851004785076611405015441616047124089938250956800",
  "regulator": "545624411053203.73045398884583072180213372155149257502745762152",
  "points": [
    [
      "11421650765628186754/9",
      "-38599643258316838007145121946/27"
    ],
    [
      "5657778683304010828/81",
      "-13359296359452931217293552370/729"
    ],
    [
      "1859392181426721484548004/45873529",
      "-2482023862685444193093304134783782266/310701411917"
    ],
    [
      "6670385051890276",
      "-208910006765036920615298"
    ],
    [
      "953233885740561876/169",
      "-121770304938003450603963594/2197"
    ],
    [
      "787993857911111269/144",
      "-9344055086171434479140719/1728"
    ],
    [
      "5476037191636420",
      "9406440863363801143966"
    ],
    [
      "154182032082462285/16",
      "41506127422013923347712539/64"
    ],
    [
      "17650888545826756",
      "2104176405239656414694302"
    ],
    [
      "22770019726862068",
      "3217519285772321257814830"
    ],
    [
      "371978685111016429303396/10569001",
      "220592085961447243696290765616630922/34359822251"
    ],
    [
      "10776150844881796",
      "-831979087315908337590818"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=6773/18: 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 12 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 16:04:49",
  "updated_at": "2026-07-01 22:48:33"
}