curve #446

y² + xy = x³ − x² − 1342601048927478697366454308878729x + 18899547549448287969118946591445127688679527385785

a-invariants: [1, -1, 0, -1342601048927478697366454308878729, 18899547549448287969118946591445127688679527385785]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 183952363112619117727161581895997057816804962203234710810829751629070049712735430
discriminant (Δ): 581791133919543550001667862947405806718057189108956771493903914454372247222058213821114575018562500
Faltings height: 17.8348
naive height: 240.4534
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 31, 53, 3823, 17109342568432627963608764579019760941917593271475015922815034559497
regulator: 586543816754491659578.067593519880199813388290721515160912493138812046098599578
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:54:59 UTC
last updated: 2026-08-30 15:54:59 UTC

Witness: 20 independent points log in to add more points →

(6764955446141926, -3182217296658307240099283)
(-41774752726744004, -1443607127631469923378773)
(20395144982752321, -24668878963199319836273)
(-4985880607469729, 5046746682262018326286177)
(12562234311879376, 2003976391231864291985317)
(62009815027246951, 13194177081020317174063417)
(350420811668446189/16, -831648405591472255002557/64)
(18073726796049496, -733279877941383395629523)
(547498844308553104/25, 781919721788556801760607/125)
(-22651368712180604, 6139155706478948292118177)
(90997381215139861, 25499653118422829418317872)
(-338087505843687776/25, -735091054485035936042297503/125)
(27838953445263721, -1760208070527138455736848)
(31889608046342596, -2917973765387700154600223)
(19117669718044216, 468370888484583743869837)
(18375899748819646, -658105757499208763050823)
(142322120153682679/4, 32172654805876026554084711/8)
(1891989726690922969/64, 1149960600610241969357246999/512)
(-10616226935504729, -5653000155283528048450823)
(555424990659573859/25, -24103322642957122895953183/125)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 19/4. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 15:54:59 UTC · history