{
  "id": 4909,
  "curve_key": "-124278996818751410064:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "2589145767057321043",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 138.80718205932212,
  "faltings_height": 9.635505583251666,
  "conductor": "156696585608668361973767378558845472",
  "bad_primes": [
    "2",
    "11",
    "13",
    "23",
    "29",
    "31",
    "37",
    "43",
    "53",
    "59",
    "67",
    "71"
  ],
  "discriminant": "-1110834805071709528154691917563284686901114748559914656448",
  "regulator": "480737771829.53200752292603642063564591630547447418862845",
  "points": [
    [
      "7668575481646542681/809061136",
      "21539800941596328766621301853/23012934952384"
    ],
    [
      "31705582054441/178929",
      "1630963930999403297402/75686967"
    ],
    [
      "1157928797041/2601",
      "4672786822102925722/132651"
    ],
    [
      "5604445451641/1089",
      "13901191859962103338/35937"
    ],
    [
      "36719374129/9",
      "7563729607514614/27"
    ],
    [
      "903718310881/1764",
      "2831784070603489847/74088"
    ],
    [
      "705887309241/64",
      "599344116505962963/512"
    ],
    [
      "1357839441370129/1065024",
      "80567970309473078306681/1099104768"
    ],
    [
      "23657208481/441",
      "109204316500523642/9261"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained by a rational 2-isogeny from curve #4764, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4764.\r\n\r\nThe source commentary attributes the original curve y^2 = x^3 - 10356583068229284172*x to Aguirre, Castañeda and Peral, High rank elliptic curves with torsion group Z/(2Z), Mathematics of Computation 73 (2004), 323–331: https://doi.org/10.1090/S0025-5718-03-01547-3.\r\n\r\nApplied the standard degree-2 isogeny with kernel generated by (0,0), followed by the rational change of coordinates dividing x by 4 and y by 8. With A = -10356583068229284172, the combined map is X = (x^2+A)/(4*x), Y = y*(x^2-A)/(8*x^2), giving y^2 = x^3 + 2589145767057321043*x.\r\n\r\nNine supplied rational points transported through this map and checked exactly on the target curve. Rational torsion C2. Same rank and conductor as the source; conductor 156696585608668361973767378558845472. Source-attributed existing curve.",
  "created_at": "2026-10-20 22:43:52",
  "updated_at": "2026-10-20 22:43:52"
}