curve #4633

y² + xy = x³ − x² + 3294088033455092464557217680072185742906x + 7154295251440899410617656893029555253206498997777225090402584

a-invariants: [1, -1, 0, 3294088033455092464557217680072185742906, 7154295251440899410617656893029555253206498997777225090402584]
c-invariants: [-158116225605844438298746448643464915659479, -6181311097244937090774367078592762038742759493098418070228300245]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 3300001758583385460319142337777169734770608439738303983614069920828533425042
discriminant (Δ): -22113749944268535066786198590742898449086654884096315729088251091092847627518434743286526828061224445339697214828335980279988
Faltings height: 22.5384
naive height: 293.7688
primes of bad reduction: 2, 3, 7, 11, 19, 67, 89, 179, 191, 257, 373, 571, 601, 661, 823, 827, 967, 1289, 2939, 3089, 3719, 4799, 8011, 15907, 75659, 318348323
regulator: 368617275324673349042.70172918315653560211213065194302572048928892601723516
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-19 03:32:21 UTC
last updated: 2026-10-19 03:32:21 UTC

Witness: 18 independent points log in to add more points →

(-690513466959110044573/4, 9604070542493819005376862132181/8)
(-172524964707315388527, 1204491931968582626911567711057)
(4773734763998476575682371877054/35715330225, 21325573082260751970018676316017873406800976572/6749661682571625)
(-99220260966710833035, 2418814838829270285290496845698)
(-23345153473067009805, 2657944928153038472504449487288)
(-10195608023874305545, 2668267262347660089526295698877)
(102259460676548083935, 2925828740859420771271189777933)
(-860955774122458110288438525083947/5334640179856, -19163566368904826082847522436992847713301563755155/12321333069170525504)
(-634463901087264140598052094792242013/5527594472666436, 942891240625839526162159078218855182598646146975151145/410964455169342814442184)
(-176971463170368195415, 1014290062623854159370516938657)
(-74040954332026669465, 2550392256064282469104019156732)
(-62827651651913711077, 2588307357903968038416337514432)
(-967298760771728824075758758361506607747604347508/5173568081224382532258247201, -18273976563984557019448135588795182126965874898007185113237082267377280/372121919758980831276919485768563388470449)
(281025549465807835143, 5502191557361505966411870395932)
(-1928768695182565945767999075872070529076976513484329531674829936426630/49854455568378023924987252992923748019852951286241, 929264927120041831903821090374018819480114858831680030975599655271294161018452913334855731399606553936518/352010782728281608529655168571222547564848462957568329335996023651672826961)
(8735426430459746152315/36, 1018657754732527280312105783077367/216)
(681363602566011766917539946069744241247967661554456604646237225/8034308882155482824702793903954483888860641, -64587439934535555662891040742183743693588044887311026574030204352954867020639647728105313423158/22773133208767987483673288116128435198751075921266729812412539311)
(-14994643662595794961415/81, 326539660213806645113247282390982/729)

Commentary

X660, T=6945/3056

last edited by Andrii Prokhorenkov at 2026-10-19 03:32:21 UTC · history