curve #3996

y² + xy + y = x³ + x² − 13202171344979051653897453095x + 583869294338556604550332937455335586810845

a-invariants: [1, 1, 1, -13202171344979051653897453095, 583869294338556604550332937455335586810845]
c-invariants: [633704224558994479387077748561, -504463070308517659113171850420005350087684441]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 142345695678637990158711155259549347070
discriminant (Δ): 355857877224989486039700387766544387389425496001600122292210470858396875000000000
Faltings height: 14.7289
naive height: 205.8641
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 41, 43, 61, 71, 89, 101, 107, 241, 439, 607, 2203, 2447, 75037
regulator: 286298674994628003.0744169016659799549771789136795620531277904005
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:41:59 UTC
last updated: 2026-10-05 04:41:59 UTC

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

(15046041377966340443/6241, -58297327206291039468852339018/493039)
(14346890095076674136947/14691889, -1707054281216648227872379352524094/56314010537)
(170291530822823, -1809411707178037990362)
(242872363599331604932693/3657709441, 56008582619219821671188690403982/221214609282239)
(95952599458062187/1444, 72744812175129262743511/54872)
(415314053547461245991323/3317644801, 180571374697435624122841437652821862/191093022892799)
(212851082197286953276543/2942628516, 13702807771664080442159828640625193/159625826478936)
(5551146326556325201183/83521321, -1194709600488396193731235957978/763301352619)
(416418199222232170939693/6135432241, 10426022287095588313380375532586482/480582272005289)
(180499858065852590864083/2713263921, -353711272237824201992456431987678/141331204380969)
(50023425757490077886890747/476145541089, 196157088671283666646354425823225368106/328556136154265937)
(157057359253512261242197/2310436489, -2577307459913236263276285380810506/111055750716763)
(931122468529649911306765807/14014144141209, 62835725890125187411019875272306408274/52462607257390528323)

Commentary

Unlisted rational-isogeny neighbor of curve#3636 (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 04:41:59 UTC · history