curve #454

y² + xy = x³ − 131235162921679918505097922631000x + 578061958908714844618861371898647409879281000000

a-invariants: [1, 0, 0, -131235162921679918505097922631000, 578061958908714844618861371898647409879281000000]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 48571732705069193627462048891253241365235087439863767107855112003052079310
discriminant (Δ): 298825468998917412750017160115596332210490894333456031803464670941710970484839291899514880000000
Faltings height: 17.2311
naive height: 233.4772
primes of bad reduction: 2, 3, 5, 13, 17, 23, 41, 1321, 5881059288089786454738776702503323780607714810733681828073617179
regulator: 70349526889012452535.0097970829042131158143485048489553615285752530908121405306
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:05 UTC
last updated: 2026-08-30 15:55:05 UTC

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

(6862235758657840, 25296868339673097611560)
(10315239775666750, 567383753741380747100500)
(5508621668158000, -149317584898839295847000)
(980617818159280, 671053941424043947025320)
(662546404414000, 701001718695675577129000)
(33117792036166450, 5705702829135643387346500)
(1038975151056130, 665457306551628033928900)
(17698112821714600, -1949078438824482256643600)
(4588031801880100, -269312259069224891846600)
(108190127928602800, -35394298866193127405811800)
(41876785154258992, -8277693740896341868236248)
(6195016754995330, -53025654583107238716230)
(6438448507060000, 2848264865655133081000)
(297693003001101520/49, -24259777600490980330448840/343)
(8050728316248880, -208146694220975479568600)
(5818510120710640, -107027310861226873165400)
(12789484226498800, -995800912668197490191000)
(19035914613968800, 2231109512317680899222200)
(19520233190737840, 2335444475839526334130600)
(5508534115689880, 149329365122860093918720)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -15/7. 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 15:55:05 UTC · history