curve #3650

y² + xy = x³ − 3635332259215271286970001417235900x + 84311290079639932822856655654964969491572101846032

a-invariants: [1, 0, 0, -3635332259215271286970001417235900, 84311290079639932822856655654964969491572101846032]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 216095858582728308893742177217643607485136038653170
discriminant (Δ): 3941612203978209437354651320619679980678243786566130660883873496471712063994279661674898329600000000
Faltings height: 18.0443
naive height: 243.4416
primes of bad reduction: 2, 3, 5, 7, 29, 31, 37, 67, 79, 139, 157, 211, 241, 257, 509, 709, 2801, 12301, 16529, 50263, 53617
regulator: 1758684180088111.3300535324698546066450750364767343833781353746552823
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:26:57 UTC
last updated: 2026-09-29 19:26:57 UTC

Witness: 15 independent points log in to add more points →

(-65190603024930516, 6652278419230975966040928)
(23998194782033304, 3300115022899654561593348)
(29391200688944304, 1689344115135726282966348)
(29420146807628304, 1680399762414432386073348)
(31584462345719304, 999624046964685922239348)
(144357933635405091/4, 2757434274236009285496159/8)
(466441332170465904, 316023929611923409666002948)
(515111910532102904, 367275981952250591129542148)
(2600084813769317091/4, 4175150443397367506982967659/8)
(-168978512601106805481864/8720209, 312720578507529989260084017473367396/25750777177)
(-485147613487664876/81, 7500916281543350892163020692/729)
(148044979510031128/9, 145306817181490729468338796/27)
(19068116877419614392/529, 3981274245413000608702763292/12167)
(3899823088292135073111/77284, 116411993534498724577730306654871/21484952)
(77445206462484645004584/19321, 21549785865935750076883820403644412/2685619)

Commentary

Specialization at t = -512/1427, u = 2/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), 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 19:26:57 UTC · history