{
  "id": 6275,
  "curve_key": "142172256:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-2961922",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 56.317649852437896,
  "faltings_height": 2.761377899344648,
  "conductor": "334010020096",
  "bad_primes": [
    "2",
    "41",
    "881"
  ],
  "discriminant": "1663032844554620764672",
  "regulator": "1677.33247833761453881946402652794027570632730391",
  "points": [
    [
      "-1681",
      "15129"
    ],
    [
      "-1312",
      "40344"
    ],
    [
      "2050",
      "50430"
    ],
    [
      "-881/49",
      "2502921/343"
    ],
    [
      "6929/4",
      "65559/8"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 5. Rank proven (independent generators plus a matching 2-descent upper bound) in M. Rubinstein's elliptic-curves-by-rank collection, file families/family_Z2.add_2026-10-24.part1.tsv.gz:\r\nhttps://github.com/mrubinst/elliptic-curves-by-rank",
  "created_at": "2026-10-31 23:35:42",
  "updated_at": "2026-10-31 23:35:42"
}