{
  "id": 219,
  "curve_key": "3674029243428660519943129:-4233416659961556177101757991695835933",
  "ainvs": [
    "1",
    "0",
    "1",
    "-76542275904763760832149",
    "4899787800875052237542378473282316"
  ],
  "rank_lower_bound": 17,
  "naive_height": 169.68999353453933,
  "faltings_height": 12.182303819386327,
  "conductor": "7123528866816955131107217283375402978917502984525303156208430",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "19139",
    "4326528036023",
    "130255667791004904223",
    "3144998910355074343693"
  ],
  "discriminant": "18328723162152475758783373844977983009407970299659748341711623258000900",
  "regulator": "37834158862449849.9148476359393670234735188524546107191970282166892065793",
  "points": [
    [
      "-285032152137",
      "-59664226043498732"
    ],
    [
      "-45981077007",
      "-91225347150091292"
    ],
    [
      "-58759497417",
      "95887962972220708"
    ],
    [
      "488170460409",
      "-289603777638798278"
    ],
    [
      "608605557828",
      "428653020829183768"
    ],
    [
      "-111700832262",
      "109799527421601418"
    ],
    [
      "-9060111282",
      "74783195378869798"
    ],
    [
      "114555611787/4",
      "418086210615852599/8"
    ],
    [
      "38448438639",
      "-44874226224542888"
    ],
    [
      "-108076640691",
      "109132194752635672"
    ],
    [
      "329936981943",
      "124747886038034548"
    ],
    [
      "-231587164254",
      "101021753503259227"
    ],
    [
      "-277722412167",
      "-68823086504141492"
    ],
    [
      "2073419014053",
      "2959722195325353418"
    ],
    [
      "-3363564171036/25",
      "-14121414026518139497/125"
    ],
    [
      "-22244479262337/121",
      "150338829342059702108/1331"
    ],
    [
      "7726866796671/25",
      "12970820812301328716/125"
    ]
  ],
  "submitter": "Edgar Costa",
  "commentary": "Specialization at T = -6286/9 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (1608, -870, -1080, -642, 1542, -558) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Seventeen independent rational points were obtained by searching the low-height Mestre quartic cover and mapping to the minimal model; rank >= 17 follows from the positive-definiteness of their 17x17 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).",
  "created_at": "2026-06-25 21:53:12",
  "updated_at": "2026-07-01 22:35:41"
}