curve #4753

y² = x³ + 41025014649039529

a-invariants: [0, 0, 0, 0, 41025014649039529]
c-invariants: [0, -35445612656770153056]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 60589865770340686157128399363506276 ☆ record for Mordell curves, rank ≥ 12
discriminant (Δ): -727078389244088233885540792362075312 ☆ record for Mordell curves, rank ≥ 12
Faltings height: 5.5601 ☆ record for Mordell curves, rank ≥ 12
naive height: 90.0291 ☆ record for Mordell curves, rank ≥ 12
primes of bad reduction: 2, 3, 25969, 164357, 9611813
regulator: 26036737538.065834850712116252578577612096412346636485744956148
submitted by: David Renshaw
submitted at: 2026-10-20 21:28:25 UTC
last updated: 2026-10-20 21:28:25 UTC

Witness: 12 independent points log in to add more points →

(-344797, 5816634)
(-344734, 7503225)
(-341550, 34365977)
(-322522, 86464641)
(-321337, 88569576)
(-319080, 92405773)
(-281340, 136953773)
(-184992, 186263821)
(-128070, 197292727)
(-62557, 201941094)
(-45897, 202307516)
(69795, 203383898)

Commentary

Mordell curve y^2 = x^3 + 41025014649039529. Source: N. D. Elkies, 'j=0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields', NMBRTHRY post, 30 Dec 2009, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6a3fad67.0912&S=.

last edited by David Renshaw at 2026-10-20 21:28:25 UTC · history