{
  "id": 1209,
  "curve_key": "18474879185094693193:-79114713689435449085632325701",
  "ainvs": [
    "1",
    "0",
    "1",
    "-384893316356139442",
    "91567955626994363782396164"
  ],
  "rank_lower_bound": 12,
  "torsion": [
    2
  ],
  "naive_height": 133.08883080399988,
  "faltings_height": 8.914452882237747,
  "conductor": "13361790818375694829477764814281226",
  "bad_primes": [
    "2",
    "13",
    "29",
    "41",
    "71",
    "73",
    "107",
    "149",
    "293",
    "857",
    "1193",
    "180241",
    "96875897"
  ],
  "discriminant": "27042412530937835487705478809524704801262020180377152",
  "regulator": "1281489280176.3484702951801664751442176085584953152223415628141",
  "points": [
    [
      "389951421",
      "880200493281"
    ],
    [
      "2113709301",
      "93388321856081"
    ],
    [
      "3486705053",
      "202822867504785"
    ],
    [
      "-15007611180808083/149499529",
      "20777008720264766285295355/1827930741083"
    ],
    [
      "3383010496/9",
      "908425017020/27"
    ],
    [
      "35550156013/784",
      "189103415461495471/21952"
    ],
    [
      "67678028958/49",
      "16068876085688672/343"
    ],
    [
      "172786853628/49",
      "70778617888624337/343"
    ],
    [
      "517528743818/1369",
      "15180374556447177/50653"
    ],
    [
      "2100355461121/6400",
      "396282340982497529/512000"
    ],
    [
      "911213947506/2401",
      "46639103872795013/117649"
    ],
    [
      "56428497118429/137641",
      "83569941252418348155/51064811"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Julian Aguirre, Alvaro Lozano-Robledo, and Juan Carlos Peral, Elliptic curves of maximal rank, Example 1.4(4) and Corollary 3.6: https://production.wordpress.uconn.edu/alozano/wp-content/uploads/sites/490/2014/01/ALP-2-23-07.pdf . Their model is y^2 = x^3 + 4510328029*x^2 + 622726581362777216*x. Submitted in its global minimal model, with 12 rational points computed independently using PARI/GP ellrank. PARI returns rank bounds [12,12] and torsion invariants [2], proving rank exactly 12 and torsion Z/2Z. The supplied points are checked by the ICARM exact independence verifier. Found through Lozano-Robledo's own blog: https://afieldguidetomath.wordpress.com/2020/04/29/birding-and-math/ .",
  "created_at": "2026-09-24 12:20:53",
  "updated_at": "2026-09-24 12:20:53"
}