curve #5680

y² + y = x³ + 849143207391785

a-invariants: [0, 0, 1, 0, 849143207391785]
c-invariants: [0, -733659731186502456]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 3438053538448260349218292587
discriminant (Δ): -311491088636950835899526526674787
Faltings height: 4.9138
naive height: 82.2736
primes of bad reduction: 3, 7, 43, 383, 32627809
regulator: 2774191462084925.102963517260565616834837051377060213650914
submitted by: jesper-petersen
submitted at: 2026-10-24 07:39:09 UTC
last updated: 2026-10-24 07:39:09 UTC

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

(-432880064615/5391684, 227985526228058497/12519490248)
(-314845259333951/33035970564, 174883804680830621193179/6004551937771512)
(154610641899919525105/977800642198416, 2118896034165447656192820218071/30575634432618597430464)
(-272376563089807143911/3118552112624004, 2355074553341602464596702814239/174152417940270654127992)
(288314144774907545641/709251386387076, 4926366190687532297083027466113/18888622631447284552824)
(-355896264254054946455/5336168381042724, 9162156789444689425217999653705/389802201632597190979368)
(1171734635152553234809/66435973907125824, 500603506955765901251096611258189/17123989733363548934456832)
(115224653220326025649271542825/1911873417276637128313476, 86394171740963975979315618458120778161688961/2643557141015236947898284402087343224)
(556455492234011787051170929/2753518877191573779600, 13785129313082209674370206670367325282967/144488102366437087447155817656000)
(249561145096264451387671819681/2163857905218086285791716, 155390475684608279363279065001662218593307923/3183047412436562426926997683212060264)

Commentary

Exact-rank-10 Mordell curve obtained by rational 3-isogeny from the Burkhardt specialization with parameters (u,v,w)=(-3,2,-4) and t=685/209, 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 07:39:10 UTC · history