{
  "id": 319,
  "curve_key": "70983282222286088736890110806923446995820745779217179382121:-18673056033031609550980836535707844686036168926500760789336296215048918683105411432710581",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-1478818379630960182018543975144238479079598870400357903794",
    "21612333371564362906227820064846376685227380006732368388851217276076518563786908263808"
  ],
  "rank_lower_bound": 15,
  "torsion": [
    2,
    2
  ],
  "naive_height": 406.5293840649556,
  "faltings_height": 31.752856890705033,
  "conductor": "50641610943684620657478317246146550331473995146991135940486156432435703726760290",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "19",
    "23",
    "29",
    "31",
    "43",
    "53",
    "59",
    "67",
    "79",
    "101",
    "107",
    "109",
    "167",
    "233",
    "281",
    "307",
    "317",
    "419",
    "571",
    "761",
    "863",
    "1021",
    "1531",
    "1543",
    "1993",
    "3299",
    "3727",
    "3821",
    "16417",
    "19273"
  ],
  "discriminant": "5193990139081890317459401622566217412295259126704665225250680699618492464914335380075351240164897155445181855895609689483201815665345724128042696272338696704023011289062500",
  "regulator": "75452645054176208.282145425803742497370186324948193253851984275183296",
  "points": [
    [
      "11183035360176737872066462162",
      "2544250119932764965168254926681650601072814"
    ],
    [
      "-14387255078284602255172684728",
      "6317468477138182570583042837408855896541364"
    ],
    [
      "19438262913631476179745098142",
      "459721721339879324467548812793427818227814"
    ],
    [
      "55823249110172914794349281022",
      "10631004522185171552847940461954446457166364"
    ],
    [
      "32572289692579628851611993897",
      "2828698563670717423033345745417611680180114"
    ],
    [
      "24588470142629085616740326022",
      "341270246931310610931397778424345615952614"
    ],
    [
      "19982025847826040867579006207",
      "202476372279358043207285606116969759878864"
    ],
    [
      "24988563558081184745544566922",
      "512203846743830750521268371377174156493614"
    ],
    [
      "19409175712041011823546618522",
      "470570514278064910180708844073966212677614"
    ],
    [
      "15343558255011251183303194179",
      "1591933177167506122828050814070561825165145"
    ],
    [
      "24354530036362110771030330897",
      "205253458293167433649155238598749814431989"
    ],
    [
      "40442125414745602621502494162",
      "5286919745747422460169589189761390362993024"
    ],
    [
      "24943293985561753095203275747",
      "494620041717645188762260334392615608575364"
    ],
    [
      "266053292109858198648954299847",
      "135869710793553829756471002267336612487527489"
    ],
    [
      "1872527673822446130016910594397",
      "2561836921226411679741258783032933091672508614"
    ]
  ],
  "submitter": "Jack Cheng",
  "commentary": "The current rank record for elliptic curves over Q with torsion Z/2Z x Z/2Z, found by Noam Elkies in 2009: rank at least 15 with the 15 published independent witnesses shown here. Source and maintained record table: https://web.math.pmf.unizg.hr/~duje/tors/z2z2.html. The leaderboard's exact Cremona/Brumer verifier re-certifies every witness and independence modulo the full rational 2-torsion subgroup. Native PARI ellglobalred/elltors independently confirms the global minimal model, torsion [2,2], complete bad-prime set, and conductor. This is a documented existing world-record curve newly added to this leaderboard, not a claim of a new world rank record.",
  "created_at": "2026-08-24 11:31:13",
  "updated_at": "2026-08-24 11:31:13"
}