curve #3998

y² + xy + y = x³ − 27523608512822747330448x + 1757513093899120516616198285497102

a-invariants: [1, 0, 1, -27523608512822747330448, 1757513093899120516616198285497102]
c-invariants: [1321133208615491871861481, -1518491313130821826169318556477288565]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 551664926630833203374698683114972985506
discriminant (Δ): 46716873234351036450295482854502951294135814155220556820726813072
Faltings height: 11.5290
naive height: 166.6216
primes of bad reduction: 2, 3, 11, 23, 47, 53, 59, 89, 157, 251, 317, 521, 7607, 9533, 55997, 95569
regulator: 57045032385808008.18910216333016948425548331669052612309016721625
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:42:48 UTC
last updated: 2026-10-05 04:42:48 UTC

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

(140887294038794/49, -1669551509974540502871/343)
(459657693664721/625, -9623205723589619754681/15625)
(201486730027061/1681, -925989536352909602637/68921)
(485630208248474/4489, -2040941007148613559291/300763)
(140214865743002/1369, -182083315440928932162/50653)
(17247910730249/169, -7460335409400588645/2197)
(93473509483493/961, -23165144498961170475/29791)
(7612555210427812682/64368529, -6461841259800052175078788518/516428708167)
(1468989065032661/14161, -7275208548279349626003/1685159)
(2119667027238961/9216, 77098427401117236495865/884736)
(1517242841047361/3136, 56002163865331757544603/175616)
(156994524195506/961, 1199456461544758819833/29791)
(7975796181397927109/82319329, -423365130488066672845977345/746883272017)

Commentary

Unlisted rational-isogeny neighbor of curve#3638 (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 04:42:48 UTC · history