curve #3995

y² + xy + y = x³ + x² − 362782262437209184474815x + 133997838633518807791458830597605797

a-invariants: [1, 1, 1, -362782262437209184474815, 133997838633518807791458830597605797]
c-invariants: [17413548596986040854791121, -115774132579490851546297824942742342169]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 1284522297763555835148164816047706661990
discriminant (Δ): -4700989739762807590934684446588261310936156133978958574852586645757920000
Faltings height: 12.6415
naive height: 175.2894
primes of bad reduction: 2, 3, 5, 11, 31, 59, 73, 191, 193, 487, 641, 691, 751, 4951, 15161, 340519
regulator: 151459265524179170.8828501662322140012386772097013162773610260699
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:41:48 UTC
last updated: 2026-10-05 04:41:48 UTC

Witness: 13 independent points log in to add more points →

(-229881284256260962313/315027001, 563560942049416499731831195874/5591414240749)
(-4947419586539801/7225, 152088968843863712898398/614125)
(-835752418814066903/2070721, 1380609530732579105330587774/2979767519)
(62150705714154883/31684, 14889509093517192197376677/5639752)
(402532241883095/169, 7854878776802735680686/2197)
(-737609894693, 16716280373268146)
(1564805287150906087/3644281, 1666633584489977024000757314/6956932429)
(12814859314843/9, 42726778700205138862/27)
(19175936652449203/3969, 2636303391063933467024422/250047)
(434527389503975/24964, 1409441598000510494337021/3944312)
(5551230552495938525/8579041, 10365997583626232786215610424/25128011089)
(15397528734640955/22201, 1537384024320701675859762/3307949)
(31136289392193143683/23551609, 160220603545094625591296454922/114295958477)

Commentary

Unlisted rational-isogeny neighbor of curve#3639 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 13: 13 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:41:48 UTC · history