curve #4756

y² + y = x³ + 1228046005329697213505

a-invariants: [0, 0, 1, 0, 1228046005329697213505]
c-invariants: [0, -1061031748604858392468536]
rank (lower bound): ≥ 14
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 651497900201089941708666232118910606092447907 ☆ record for Mordell curves, rank ≥ 14
discriminant (Δ): -651497900201089941708666232118910606092447907 ☆ record for Mordell curves, rank ≥ 14
Faltings height: 7.2779 ☆ record for Mordell curves, rank ≥ 14
naive height: 110.6426 ☆ record for Mordell curves, rank ≥ 14
primes of bad reduction: 3, 293, 7459, 170389, 4397072149
regulator: 97603961339744.4970150705940727897781775456582408801955852165813574
submitted by: David Renshaw
submitted at: 2026-10-20 21:28:37 UTC
last updated: 2026-10-20 21:28:37 UTC

Witness: 14 independent points log in to add more points →

(-10708499, 285542477)
(-8815247, 23302911438)
(-32362295/4, 211426880527/8)
(-6756749, 30324506652)
(-6480707, 30916971746)
(-1343057, 35008904528)
(-230045, 35043313644)
(481801, 35045083062)
(5388601, 37209070457)
(7950751, 41601061152)
(27821173, 150871221333)
(71514823, 605789818983)
(328135537/4, 5950622644813/8)
(1345705555, 49365686078019)

Commentary

Mordell curve y^2 = x^3 + 78594944341100621664336. 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 21:28:37 UTC · history