{
  "id": 4184,
  "curve_key": "48931317086828601:-10925283931640662614488685",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-1019402439308929",
    "12645004805360636001409"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "naive_height": 115.30624392333446,
  "faltings_height": 7.470683183815275,
  "conductor": "638558705203558998344002920820447810861615354",
  "bad_primes": [
    "2",
    "15233",
    "11817766043",
    "1773576820123072327762918950983"
  ],
  "discriminant": "-1277117410407117996688005841640895621723230708",
  "regulator": "50264258944901.41876483176115798707785283179065239534857355604567163576",
  "points": [
    [
      "18375848",
      "10836353017"
    ],
    [
      "-2495214974/81",
      "88743218573861/729"
    ],
    [
      "1371565106",
      "50781104070165"
    ],
    [
      "73246295/4",
      "-87072560481/8"
    ],
    [
      "-34805268",
      "-77198171075"
    ],
    [
      "38265764",
      "172225317943"
    ],
    [
      "475680384/25",
      "-1464977395133/125"
    ],
    [
      "-19562603288342797/1533583921",
      "-9220415050164788363962635/60056679930281"
    ],
    [
      "2927203630953/157609",
      "-681741570823967701/62570773"
    ],
    [
      "218839048255/484",
      "-102127293785736751/10648"
    ],
    [
      "834265445429/44944",
      "-103747406774419153/9528128"
    ],
    [
      "15561697996/289",
      "1657816522933069/4913"
    ],
    [
      "587882384356/30625",
      "65594728908191979/5359375"
    ],
    [
      "69288795",
      "524048347088"
    ],
    [
      "4009496739529269/215854864",
      "-34558856836081578607915/3171339661888"
    ],
    [
      "3192650291/169",
      "-24985825959024/2197"
    ]
  ],
  "submitter": "Andrii Prokhorenkov",
  "history": [],
  "commentary": "Independently constructed eight-point plane cubic pencil 1005a6f72cab52f0 at t=-114/1709. Rank exactly 16: exact local 2-descent characters certify 16 independent rational points, and unconditional PARI algebraic 2-descent gives upper bound 16. The displayed quadratic base change embeds the curve in the independently derived K3 family recorded in companion metadata.",
  "created_at": "2026-10-11 23:04:01",
  "updated_at": "2026-10-11 23:04:01"
}