{
  "id": 318,
  "curve_key": "5155427047880997844813598639961401268222342979321:-42316075968944056233625131165228935370252001883517138331286420622090154381",
  "ainvs": [
    "1",
    "0",
    "1",
    "-107404730164187455100283304999195859754632145403",
    "48976939778870435455584633601954124477725966045276826839333933204370006"
  ],
  "rank_lower_bound": 15,
  "torsion": [
    3
  ],
  "naive_height": 339.0625875119425,
  "faltings_height": 26.310587474280016,
  "conductor": "1100952874635444134388803603028092741241085590888213307674333870",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "19",
    "29",
    "71",
    "173",
    "227",
    "367",
    "467",
    "907",
    "1039",
    "2203",
    "377239968949",
    "1955627779763108243655451"
  ],
  "discriminant": "-956960149635890553026355922064884560686815442397416962941379345991616974205586950201323526133577691235859622088389207804542013981650254781250000",
  "regulator": "10832019999044639.559291391070690516529366423614088218275519628150461",
  "points": [
    [
      "-422511571742285267531345",
      "-137592127148862829770221635303960828"
    ],
    [
      "-441007351163624462047670",
      "-102824147331859218513919290250298203"
    ],
    [
      "156085384573658420738746",
      "189776933096927384678335033331045984"
    ],
    [
      "-460838553366765681840845",
      "-24574287549951174429604416824455828"
    ],
    [
      "4176134452254148300203630",
      "8510736658900069929975759728364918172"
    ],
    [
      "-359187750061494853894325",
      "-203013531407261210327764965678707548"
    ],
    [
      "-460188579405761971770845",
      "-30783285810055207778934556373103328"
    ],
    [
      "646948036481989988599630",
      "500266360594011622677409610820423797"
    ],
    [
      "339470884962917809004155",
      "227237649388778691045137980486286672"
    ],
    [
      "-126915798498020139215720",
      "-246097512030324095711517097854917203"
    ],
    [
      "1045627763773449027515596",
      "1039180222036999853815782266061861389"
    ],
    [
      "17730115810583176883376655",
      "74644027306445943530330235569361589172"
    ],
    [
      "256495517365884675394222",
      "195711357157735899784726812610011893"
    ],
    [
      "-142791994651726392782285",
      "-247794290462672857095557195444679328"
    ],
    [
      "1113480161648803705205155",
      "1144516928010965576054744619118736672"
    ]
  ],
  "submitter": "Jack Cheng",
  "commentary": "Rank-15 curve with rational 3-torsion found by Noam Elkies and Zev Klagsbrun (2020); 15 published independent witnesses, re-certified here with the leaderboard's exact Cremona/Brumer quadratic-character verifier. At preflight the live database's only Z/3 curve had rank lower bound 13, so this raises that torsion class to rank at least 15. Source: https://web.math.pmf.unizg.hr/~duje/tors/z3.html",
  "created_at": "2026-08-24 11:02:50",
  "updated_at": "2026-08-24 11:02:50"
}