curve #3968

y² + xy = x³ + x² − 1068263567637287110796687941x − 4158220470450085548213314417698208979635

a-invariants: [1, 1, 0, -1068263567637287110796687941, -4158220470450085548213314417698208979635]
c-invariants: [51276651246589781318241021193, 3592702486468489338771954233531365750745755]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1321043258541830079458759813032686
discriminant (Δ): 70552042644207985438560994076801910596606186261102474837966356990503199580610572544
Faltings height: 14.5889
naive height: 198.3211
primes of bad reduction: 2, 3, 7, 13, 17, 43, 101, 131, 179, 509, 587, 1367, 2141, 13217, 120917
regulator: 249811848849671.7137689628501406653838830920108058812197360
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:36:21 UTC
last updated: 2026-10-05 04:36:21 UTC

Witness: 10 independent points log in to add more points →

(49563369211885905146/49729, -348743945339319266212674356539/11089567)
(-16353069938539745542334271029831/1628113262236373056, 154880953943729073627275233202978460381420061583/2077432497077173200460791296)
(1544103711018156298672139/23665207225, -1643968837321130290538921963628969058/3640537153457875)
(83600137357801813026490/2719201, -24171869577616045988823356306936485/4483962449)
(-86047340361524465999/6471936, -1444256844509680425220694779777/16464605184)
(1061330483836428125551786/24043913721, 694447437978297787476347892978812767/3728273305491981)
(98947801358600335321665398554/109624773934225, -31104489694034515291987822518103302637219837/1147791663674513581625)
(9443196242910326862740025139319151241210/88933484003888866055258689, 871421817816451509150451831161431456243869927031939359308485/838683230665175189221964334966806025887)
(34840459933731240790318953541932641/434072945063915982400, -5910882482298535088822238273440940679166503722824749/9043656884783059650739375168000)
(31405622457658425136215080315/643835257940281, 3994865105285018833076739848442772252297090/16336617245968217528429)

Commentary

Unlisted rational-isogeny neighbor of curve#3698 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 10: 10 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 04:36:21 UTC · history