{
  "id": 6215,
  "curve_key": "935653405156690878256:-28620189416042785782298165290200",
  "ainvs": [
    "0",
    "-1",
    "1",
    "-19492779274097726630",
    "33125219238028595154211007592"
  ],
  "rank_lower_bound": 6,
  "torsion": [
    5
  ],
  "family": null,
  "families": [],
  "naive_height": 144.86333036447635,
  "faltings_height": 9.33537910425089,
  "conductor": "475644963948942997",
  "bad_primes": [
    "11",
    "89",
    "109",
    "281",
    "701",
    "1607",
    "14081"
  ],
  "discriminant": "28093925619469856865522038767640061031447253797",
  "regulator": "2280197.3306330398676630070849603073224781269532635",
  "points": [
    [
      "7500728655083176249556/2942573144449",
      "-74610654858821821354038497/5047669373975803457"
    ],
    [
      "2549140434",
      "8985631295"
    ],
    [
      "25268263767",
      "-3959038747959973"
    ],
    [
      "-1067837158784/225",
      "-462009527047808936/3375"
    ],
    [
      "208266674005557878625/81288852544",
      "-26409067547391971214427287921/23176427326524928"
    ],
    [
      "1034664909609337916109/405833976601",
      "-9961638803674197865477353330/258536940627643651"
    ]
  ],
  "submitter": "Lewis Combes",
  "history": [],
  "commentary": "E has torsion Z/5Z and rank exactly 6. It is the Z/5 Tate normal form y² + (1−t)xy − ty = x³ − tx² at t = −89/175163.\r\nIt was found in a sweep of the Kubert Z/5 family over t = a/b with max(|a|,|b|) ≤ 600000 (4.4×10¹¹ curves): a Mestre–Nagao screen in C, followed by a 2-Selmer rank computation and a 2-descent with point search in Magma. \r\nRank 6 is certified by six independent points (Magma CertifyPoints) and 2-Selmer rank 6. It was confirmed independently in PARI/GP: ellrank [6,6], regulator 2280197.331, torsion Z/5.\r\n\r\nSearch performed by Claude code Opus 5.5.",
  "created_at": "2026-10-30 20:58:59",
  "updated_at": "2026-10-30 20:58:59"
}