curve #5750

y² = x³ − 51827014562201316

a-invariants: [0, 0, 0, 0, -51827014562201316]
c-invariants: [0, 44778540581741937024]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 33160980721365768720631700149776
discriminant (Δ): -1160369037402030979072344451640961792
Faltings height: 5.5990
naive height: 90.4965
primes of bad reduction: 2, 3, 2256613, 212654879
regulator: 4997507963083164.70235097126506288169673256220635708199363028
submitted by: jesper-petersen
submitted at: 2026-10-24 18:43:26 UTC
last updated: 2026-10-24 18:43:26 UTC

Witness: 11 independent points log in to add more points →

(1788607659077506/1624170601, 74161416045780386030510/65455699390901)
(1796661414305322/1556854849, 74860135266432833695458/61428821776993)
(7677548129529445/65334889, 672719186901534509376539/528101907787)
(7723640554384045/1275704089, -678707057718357362295911/45564322946813)
(215823221839656085/351218613769, 88360616592209587417574291/208145145608218853)
(1559562144011296145106/84719120081401, 61588868185518182002990192252050/779780112246119142899)
(23904837444237737145/61547351324944, 39710771544231288115574200659/482852019182666568128)
(23924392582267480729/61440953511184, 40902981943314510116892600515/481600490348762978752)
(502617196592622465/200476480516, -355746543790694169019048767/89762542245116936)
(90305489485387088433330/19085443578721, -27137586857289422416206965588924118/83378367422479066769)
(186841721391894475897/130500875995204, 2531289665637077140649022241357/1490802596104659944392)

Commentary

The rationally 3-isogenous partner of the Mordell curve constructed from the Burkhardt quartic using parameters (u,v,w)=(-9/9,-12/9,-10/9) and specialization t=191/422. PARI/GP 2.17.4 independently returned matching computational rank bounds [11,11].

last edited by jesper-petersen at 2026-10-24 18:43:26 UTC · history