{
  "id": 2315,
  "curve_key": "786646069844699988872656:-495421189220315719294107792002534080",
  "ainvs": [
    "0",
    "1",
    "0",
    "-16388459788431249768180",
    "573404154184717410734999527857984"
  ],
  "rank_lower_bound": 14,
  "torsion": [
    2
  ],
  "naive_height": 165.06619613844882,
  "faltings_height": 11.782291928753029,
  "conductor": "12853118643121903611159945035499392612472",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "17",
    "29",
    "59",
    "71",
    "73",
    "79",
    "109",
    "163",
    "13306129",
    "45371593",
    "54443929"
  ],
  "discriminant": "139666611043087056720755188897624419869125934752578076121913814870272",
  "regulator": "8833567633192772.94582791106285634093459235715363923942517567056843",
  "points": [
    [
      "319676299710",
      "167341153930441722"
    ],
    [
      "-19818388858",
      "29839788596099874"
    ],
    [
      "709956150918",
      "588882846967431390"
    ],
    [
      "-400925437359234/2809",
      "329699039874202092066/148877"
    ],
    [
      "-509365626379/4",
      "195204358330507239/8"
    ],
    [
      "-68649249161046/529",
      "276036935034835780722/12167"
    ],
    [
      "33808314006906/169",
      "159956248766856664374/2197"
    ],
    [
      "158736455818048/9",
      "1999878454176200578232/27"
    ],
    [
      "8767645419505146/1369",
      "820801536534807883376454/50653"
    ],
    [
      "8206459694805588/73441",
      "233267797729074510843840/19902511"
    ],
    [
      "26307174391240686/1274641",
      "22476958728235001021549574/1439069689"
    ],
    [
      "721271790628013274/6027025",
      "267178927406490005261475582/14796346375"
    ],
    [
      "10525125867356841/15376",
      "1061724665191682198659491/1906624"
    ],
    [
      "-22976111690685530977146/271752647401",
      "5214012095773271554235730572667738/141664383337493899"
    ]
  ],
  "submitter": "Steps Unbounded",
  "history": [],
  "commentary": "2-isogenous partner of curve#2177 (found by Alvaro Lozano-Robledo). The rational 2-isogeny with kernel generated by the 2-torsion point (0,0) of y^2 = x(x^2 + a x + b) maps it to y^2 = x(x^2 - 2a x + a^2 - 4b); this is the minimal model of that image. The 14 points listed are the images of curve#2177's generators under the isogeny, saturated with PARI (ellsaturation, primes up to 200) and LLL-reduced. PARI's 2-descent (ellrank) gives the upper bound 14, so the rank is exactly 14. Torsion Z/2Z. curve#2177 is the fibre (a,b,c,u) = (19,139,196,12) of Kihara's Z/2Z construction.",
  "created_at": "2026-09-26 19:53:55",
  "updated_at": "2026-09-26 19:53:55"
}