{
  "id": 6272,
  "curve_key": "15357360:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-319945",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 49.6413161877348,
  "faltings_height": 2.2050167606193902,
  "conductor": "3275673696800",
  "bad_primes": [
    "2",
    "5",
    "61",
    "1049"
  ],
  "discriminant": "2096070841845352000",
  "regulator": "2024.62092976310497185238302313778716716312557525",
  "points": [
    [
      "-245",
      "7980"
    ],
    [
      "-61",
      "4392"
    ],
    [
      "605",
      "5280"
    ],
    [
      "-1220/9",
      "172630/27"
    ],
    [
      "14945/16",
      "1453935/64"
    ]
  ],
  "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 23:34:15",
  "updated_at": "2026-10-31 23:34:15"
}