curve #453

y² + xy = x³ − 947722297710433437416491166575338x + 10741271823987029219560000495736411788978103849892

a-invariants: [1, 0, 0, -947722297710433437416491166575338, 10741271823987029219560000495736411788978103849892]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 13084625393869787313650703588177493604700881735528116368457036420717249161149679450
discriminant (Δ): 4636299657612563166681253329091306846826122114695724901421139822853342167905934630275298181345280000
Faltings height: 17.8834
naive height: 239.4084
primes of bad reduction: 2, 3, 5, 13, 43, 53, 79, 521, 189757, 18934786237, 60273449057849, 329976530182267181, 23279935841954019329597
regulator: 1429616378236788333106.93030005646862421948609932056411497257016340596683829822
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:04 UTC
last updated: 2026-08-30 15:55:04 UTC

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

(-883163106230566956/25, -45893935868852866927232082/125)
(-13404007074981768, 4586533014423937630152774)
(-29187445302236508, 3679385378625613483479894)
(9703420698107892, -1568044189600465761158586)
(14636943133937652, 73038813676268703073734)
(-9131926324219587468/841, -108391943459126091311372599554/24389)
(364695865829955012, -219478432608439580719420746)
(29141772119995602, -2805602827052322342979116)
(22074431944533276, -759790577070636912242970)
(-3738862885412172, 3772586598169086742736262)
(-8414127190500588, 4256739167268168764394054)
(14651572852284942, 29698099457936827787544)
(96127560077743092, -28423696307206497980400186)
(8055223485925332, 1905212105488130066426694)
(-25172864248623648, 4318192896687876152747514)
(2814750489621228, 2845341943746707529664110)
(-5530367090280111468/841, 99634587561235636514517988446/24389)
(1656723953981623044/25, 1932298619878530161696337918/125)
(88252510990242243/4, -6047695630033812578474373/8)
(21203413701522132, -423149363319236182950426)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -13/9. 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 15:55:05 UTC · history