curve #3643

y² + xy + y = x³ + x² − 86374399929534275305324462913x + 9855469784322937076113338981697664456740031

a-invariants: [1, 1, 1, -86374399929534275305324462913, 9855469784322937076113338981697664456740031]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 84518160226589403767572961475020918466832050
discriminant (Δ): -718721163078248688930564988199526733848361630919573154120071187972966941730914304000000
Faltings height: 15.4849
naive height: 211.5163
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 47, 67, 71, 79, 97, 167, 173, 211, 313, 367, 431, 743, 1621, 4643, 5791
regulator: 7126689268899172.85099538632183730278605038052372059139001175436114
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:25:36 UTC
last updated: 2026-09-29 19:25:36 UTC

Witness: 14 independent points log in to add more points →

(-238644710244435, 4108179458614153564042)
(-188579236247435, 4408810495291602930392)
(173687088201055, 304968260589798877472)
(383895309535815, 5768322588638028403792)
(2930434854135415/9, 108845376563237961664064/27)
(8307885531825231/49, 99870495682830193816384/343)
(13216661435690378215/729, 48042537208162462928295096256/19683)
(49779750086093919551/297025, 47740671665086070458331478464/161878625)
(116441217397012396295/273529, 1013893524637339840183922511424/143055667)
(575473705287041346335/3996001, 5072480924666665759194444175168/7988005999)
(590497480367429852135/2968729, 3774508988974582188978953920064/5115120067)
(174805508556681069035595/1068439969, 11225569723415773685968794742963316/34924097266703)
(258876945727843615189215/2778238681, 236855560385929576557970115798304368/146438182636829)
(82798079153146389433990215/62374563001, 736336637403837876527580108926383702208/15577984734936749)

Commentary

Specialization at T = 595/198 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:25:36 UTC · history