curve #5595

y² + y = x³ + 752829892263

a-invariants: [0, 0, 1, 0, 752829892263]
c-invariants: [0, -650445026915448]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 244837229767961732153553843 ☆ record for Mordell curves, rank ≥ 10
discriminant (Δ): -244837229767961732153553843 ☆ record for Mordell curves, rank ≥ 10
Faltings height: 3.7424 ☆ record for Mordell curves, rank ≥ 10
naive height: 68.2174 ☆ record for Mordell curves, rank ≥ 10
primes of bad reduction: 3, 283, 10640705191
regulator: 12614262728081.69263292029988627474180475272364969645887713
submitted by: jesper-petersen
submitted at: 2026-10-23 21:40:30 UTC
last updated: 2026-10-23 21:40:30 UTC

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

(-46313722529223/5699344036, 200073707530023865833/430266278653784)
(-658909402503/102292996, 720929688190981703/1034591361544)
(-35029616073623/11600582436, 1064086894648844039177/1249452331851816)
(-391917915423/149915536, 1573629509951020593/1835565822784)
(-216372525207/177369124, 2047108869590738679/2362201993432)
(1291611574377/13524294436, 1364647092674864538673/1572794297140184)
(51523685077593/19763298724, 2438872196548851425881/2778364061217368)
(1719286391649/405619600, 7437911110301520143/8169178744000)
(2232875906217/455139556, 9061542768557156913/9709947287704)
(2374155791649/468289600, 9523284499483654607/10133786944000)

Commentary

The rational 3-isogenous partner of our accepted rank-10 Mordell curve obtained from the Burkhardt quartic with parameters (u,v,w) = (7,-2,3), specialized at t = -208/211. Constructed using the rational 3-isogeny y² = x³+B → y² = x³−27B. Rank is exactly 10 because the original curve has certified exact rank 10 and isogenous elliptic curves have equal Mordell–Weil rank.

last edited by jesper-petersen at 2026-10-23 21:40:30 UTC · history