curve #465

y² + xy + y = x³ − x² − 1081299566142772205522196792975080x + 13444054816725382360652400584113824513310769285547

a-invariants: [1, -1, 1, -1081299566142772205522196792975080, 13444054816725382360652400584113824513310769285547]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 552136283056792471612235047565760868415703923468362182316699199502637519650
discriminant (Δ): 2832147139862193411730073077711106912122893965070091404383155925798452373209165911247341766656000000
Faltings height: 17.8741
naive height: 239.8040
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 47, 18311, 126097, 2098363, 491413837938047, 237384597778199107130020958230954937
regulator: 68655810722195745653.6958697703802531988591140070205372728375888837505394785991
submitted by: Bhavik Mehta
submitted at: 2026-08-30 20:01:09 UTC
last updated: 2026-08-30 20:01:09 UTC

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

(21465017867702669, 351969769514110501391865)
(380532689182040369, 233891125788825341164695165)
(7866854607363869, -2329052929871642639426535)
(22044170560312589, -565656498803084714175495)
(-36719844392393331, -1907384479022208864582535)
(28286309939395053, 2343158188532423563967321)
(42690338089442589, 6714510856222717365524505)
(531421511023887749/25, -31627897373751726068186667/125)
(-329245570656194699/9, 54283381878156595171181635/27)
(794568163704975101/49, -143049151521797172622493025/343)
(22889852294169549, 828431554905451092565625)
(-24508707674645011, -5022300145365397618045095)
(-37810377567782131, -523212950167446699350535)
(69466345758813549, 16539197684665564599517625)
(15962683507095299, -501018758633353554517125)
(15385182101222765, -670668023859907374597927)
(81887547763773869, 21771582533835410351313465)
(21148544439483149, -187244728200388638810375)
(21933473706366669, -528294285584731517664135)
(24267918308505989, 1222830933960167615332905)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -44/7. 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 20:01:09 UTC · history