curve #318

y² + xy + y = x³ − 107404730164187455100283304999195859754632145403x + 48976939778870435455584633601954124477725966045276826839333933204370006

a-invariants: [1, 0, 1, -107404730164187455100283304999195859754632145403, 48976939778870435455584633601954124477725966045276826839333933204370006]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/3ℤ
conductor (N): 1100952874635444134388803603028092741241085590888213307674333870
naive height: 339.0626
Faltings height: 26.3106
discriminant (Δ): -956960149635890553026355922064884560686815442397416962941379345991616974205586950201323526133577691235859622088389207804542013981650254781250000
primes of bad reduction: 2, 3, 5, 7, 13, 19, 29, 71, 173, 227, 367, 467, 907, 1039, 2203, 377239968949, 1955627779763108243655451
regulator: 10832019999044639.559291391070690516529366423614088218275519628150461
submitted by: Jack Cheng
submitted at: 2026-08-24 11:02:50 UTC
last updated: 2026-08-24 11:02:50 UTC

Witness: 15 independent 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)

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

last edited by Jack Cheng at 2026-08-24 11:02:51 UTC · history