curve #5681

y² + y = x³ − 35352589326315

a-invariants: [0, 0, 1, 0, -35352589326315]
c-invariants: [0, 30544637177935944]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 3672898007730799119448602321
discriminant (Δ): -539916007136427470558944541187
Faltings height: 4.3840
naive height: 75.9159
primes of bad reduction: 3, 7, 2885925659291
regulator: 3061070604506797.867549394859415010958739598658131694521061
submitted by: jesper-petersen
submitted at: 2026-10-24 07:40:44 UTC
last updated: 2026-10-24 07:40:44 UTC

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

(81604371605425/586705284, -732315585050433368045/14211175389048)
(1156805376704017721/2292147552256, 1244029613358160264489112173/3470274719754747904)
(42630151541724941808879714305537/1011996953437890546464755264, 202071160991430598480030723825206870681723926017/32193545434966675268411717132619687142912)
(425843414053455263604598591062913/11414195956047153455012610576, 4964999948196318902833835673277769418664753508063/1219461198384456752640289424045440646026176)
(1637560360401728393/11305236130276, 2083316379485714391038567143/38011889376966381976)
(3635371526165233873/141161009796, 6931435742825733257144720347/53036167634494344)
(493521445795844825/14929599565456, -50605107787083538329674819/57686240887372391104)
(94850435732322264121/2416311352880100, -595470765143625671613528131831/118776176699060666799000)
(6551196505686885235685273/197914536149288935876, 2664570176766304961145744130217313673/2784303281538019655390507880376)
(161307579178941882189556545097/3750034854097208084106384, 48300061762784728478153932306084530299458459/7261945016377678373920006542777622848)

Commentary

Exact-rank-10 Mordell curve obtained by rational 3-isogeny from the Burkhardt specialization with parameters (u,v,w)=(2,-1,1) and t=958/647, discovered in October 2026. Exact rank [10,10] independently verified with PARI/GP 2.17.4 using ten rational points checked exactly on the minimal model.

last edited by jesper-petersen at 2026-10-24 07:40:45 UTC · history