{
  "id": 512,
  "curve_key": "1020350301162180893637262171877015190804429940432801:-32381230918076433179053477000307316147517258638290062159469098290428931192049",
  "ainvs": [
    "1",
    "0",
    "0",
    "-21257297940878768617442961914104483141758957092350",
    "37478276525551427290571152164099380097506849748144449472160083430305579332"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 352.355957230635,
  "faltings_height": 27.209737800531833,
  "conductor": "53408729173037795322621188411209581292325482397255882514654677241857822588089265070",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "19",
    "29",
    "71",
    "107",
    "127",
    "151",
    "277",
    "353",
    "541",
    "677",
    "947",
    "2671",
    "2969",
    "4931",
    "14557",
    "38767",
    "112589",
    "267167",
    "1613609",
    "2149909",
    "10787047"
  ],
  "discriminant": "7961585506922791313004835825027915933572411635320264996015765046933489091680934177397242361796970688822027713568669995291592829778587317574400000000",
  "regulator": "109305095885381469157.732721802577654106778872879369878438095695310221309",
  "points": [
    [
      "144322296891428228514254",
      "-5866291890058987393338787410137463652"
    ],
    [
      "243297607518051195849944584/81",
      "621834818036347610891936400713350168162/729"
    ],
    [
      "2418243784716276707426198",
      "-463252619141756858588134051209935434"
    ],
    [
      "2971888356275556171071804",
      "742991301155567671971495402215576498"
    ],
    [
      "3209895184957695125868302",
      "1522333488980749260389806165434083420"
    ],
    [
      "274693856958183488086455692/121",
      "-1276754549857430641112298224251242080330/1331"
    ],
    [
      "-2443386461187350389609846",
      "8650474994143171672214724517122467798"
    ],
    [
      "-714519374393459885836546",
      "7232028657621575159832711855127586348"
    ],
    [
      "20957742447150970905312194",
      "93793277917057420840673173682326534088"
    ],
    [
      "-25706815044539377760499284/9",
      "233659185026329304285137763537051746166/27"
    ],
    [
      "1563493829588025065892078332/2209",
      "-495609729124139030974577158109421395729042/103823"
    ],
    [
      "8482616729370666572110604",
      "-21622341264788175247998099341966287102"
    ],
    [
      "226779952527906388119748665972764/8826841",
      "-3363521406212902365968820368321702188554594714922/26224544611"
    ],
    [
      "2976726130020675050225624",
      "-759993586979651869957243448587267342"
    ],
    [
      "79852456772496183148130204",
      "712399061055359560129736514065286738098"
    ],
    [
      "5784467548400663979629668143386296/1949840649",
      "-62370751970463885273312018846642915550986125759186/86099113537893"
    ],
    [
      "694241932243774094330972076884756/30880249",
      "-17934210402735194686146925905843828576900137884366/171601543693"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 2). Fiber t=-27365/3567 on branch u=29/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).",
  "created_at": "2026-09-02 13:42:09",
  "updated_at": "2026-09-02 13:42:09"
}