curve #355

y² = x³ − 252855311567215820639960610648737067x + 49319408860526775041104717131304690483461209453955226

a-invariants: [0, 0, 0, -252855311567215820639960610648737067, 49319408860526775041104717131304690483461209453955226]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 2160288978585670445440893546206560240112122473979420022281459005989256431760
naive height: 256.1835
Faltings height: 19.2025
discriminant (Δ): -16142003783445781808920938206778271203651911584288646693594818011146385545148748364251510614775307220480000
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 313, 489539, 1439239, 7617780311, 4815998457419707882431488757218133251231
regulator: 60699467860927856405.7857195187468274002341485131977537494878518443962250770842
submitted by: 7fff-zip
submitted at: 2026-08-25 14:10:27 UTC
last updated: 2026-08-25 14:10:27 UTC

Witness: 20 independent points

(-2019638822022837268554628060527069249/19366699711524400225, 23271220611078251714252322519158890577410146068310651374/85228221921363254612537284625)
(237292716256906007, 51769912522766509079198190)
(75719202682312571307917/57121, -19515390232494012363627781135587840/13651919)
(6241929306731715293268837/7946761, 12972095229065046901674305078832534400/22401919259)
(41515401984098717524106/143641, 1063660727220632857541293193615490/54439939)
(2582123612099476443072143/66049, 4148869687928413151001662993884523730/16974593)
(66070813229272491636167/160801, 7842720084745284093661417461567150/64481201)
(159561026024221941854705/440896, -21131026401576460665576265096310199/292754944)
(694236840396406578437/1681, 8534154877690388294740269837960/68921)
(17832388216180302488263277/50537881, -22809133813732305932795550052228223400/359273796029)
(510503669931442179383/1849, 1870633910470598662424382494550/79507)
(4767314810716245034493/18769, 98963993966545053367132154883840/2571353)
(16102269042459691313/64, 20658837558856555451679881175/512)
(512190929896341794825/3364, 23372884323810767836511458023987/195112)
(13181425042723441447713818/51509329, -13688697532112984552723148714721654050/369682454233)
(2351806560534061093/9, 890880483732373169998066720/27)
(21883919004732365062/81, 19637444545642881553756820660/729)
(-20373664434769795, -233372109091778885949247296)
(71086902629927039285/289, 220028355418023356251758269832/4913)
(-88338389965092882328164697/220908769, -965596571633535865831787784518349081810/3283367033647)

Commentary

Found in the Mestre-family search; the listed 20 rational points are independently certified (roots=[0, 239, 3790, 4812, 4879, 5762], T=13522/7).

last edited by 7fff-zip at 2026-08-25 14:10:28 UTC · history