{
  "id": 805,
  "curve_key": "1806924079328673681:2428901248101438190522196295",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-37644251652680702",
    "-2811228287002638696230515"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    2
  ],
  "naive_height": 126.11447300958079,
  "faltings_height": 7.897925169939746,
  "conductor": "47048397659668784440854",
  "bad_primes": [
    "2",
    "3",
    "13",
    "47",
    "241",
    "277",
    "1069",
    "29231",
    "43633"
  ],
  "discriminant": "884733417110354735864749451041363483164672",
  "regulator": "25876717.32754379321327557944505826315227436346031821",
  "points": [
    [
      "250293508167/289",
      "-121761303683699563/4913"
    ],
    [
      "-31838299014453551/284225881",
      "196047071372177418825/4791764127779"
    ],
    [
      "-854752384229568023/7630546609",
      "37747342484339569376219/666551137935977"
    ],
    [
      "-268953270375/2401",
      "6610479590611/117649"
    ],
    [
      "65809660710585/66049",
      "522866676096454979747/16974593"
    ],
    [
      "-2845131514627037435799/25398949029169",
      "5085234125551043504760156955/128004023183417088553"
    ],
    [
      "-377232573021995175/3367596961",
      "14659547929779803327885/195425019243791"
    ]
  ],
  "submitter": "benvchurch",
  "history": [],
  "commentary": "Found on 2026-09-19 by searching a second elliptic fibration of the D=26 K3 companion at moduli parameter j=4. The source equation is Y^2=X^3+t^3(352836-552123t)X-t^5(323554257+158153580t+729t^2). Setting u=X/t^2 gives a rational-2-torsion pencil; this curve is its fiber u=225/4 (scaled parameter v=-75/4). Candidates were selected by a Mestre-Nagao prime-sum sieve through 251 on a reduced-rational box |numerator(v)|<=256, denominator(v)<=8, then examined in Magma. A GRH-assisted descent found the seven submitted points. In a fresh process with rigorous Minkowski class-group bounds, exact point identities and IsLinearlyIndependent verified their independence modulo torsion, proving rank >=7 unconditionally. The computed upper bound 7 remains GRH-conditional; no saturated-basis claim is made. Magma also verified global minimality, torsion Z/2, and conductor 47048397659668784440854. The exact map back to the source K3 is t=-(x+112019271)/1296, X=(225/4)t^2, Y=(2y+x+1)t^2/3456. ",
  "created_at": "2026-09-19 19:09:29",
  "updated_at": "2026-09-19 19:09:29"
}