curve #464

y² + xy = x³ − 35401605264465213409496277503084042915x + 80481020268953308189156706852613645754737587637675825921

a-invariants: [1, 0, 0, -35401605264465213409496277503084042915, 80481020268953308189156706852613645754737587637675825921]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 44513751878116744809773860414183449796086120374745760991466219808902636877123326
discriminant (Δ): 41397473566786685512109309127918766973738082017784807821951030869582035307344006094470887808377159726211356147712
Faltings height: 20.4345
naive height: 270.9931
primes of bad reduction: 2, 3, 11, 13, 17, 29, 37, 43, 71, 83, 103, 6779, 9119567, 17853897615847, 62054470788771975296126013819524170351
regulator: 114159273959719202761.126589089576499205211515498368367752389840490127554721944
submitted by: Bhavik Mehta
submitted at: 2026-08-30 20:00:57 UTC
last updated: 2026-08-30 20:00:57 UTC

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

(4190823300449749966, -2392190924489904348627874487)
(4369659586453879720, -3036775996236006904733473049)
(718682251231985614, 7443770367259922052804288841)
(3682166675910629194, 224764103909461407550790101)
(20767603268493205126, -91116428930298992861057226239)
(79482669854674265086/25, -31835217305439196881434007931/125)
(-6628662597856198898, 4887575943667654893915617929)
(-4264437845674362926, -12405576527360541106812564335)
(3833187437122701088, 1049906695094716923748723009)
(3699772208237333230, -383132032305142354899113879)
(2967800786192471374, 1247408223725656108811532265)
(3029407129826290798, -1018305369778353646698559511)
(3672629762602663150, 35075899180128590712246889)
(2996304785754007444, 1143404147540295320516402851)
(3191464387981609846, -67398747914465660495932559)
(2075600318202276010, 3992914407127106993376626209)
(1005525742642129438, 6774988105968135644141239753)
(4180199395378808518, 2353791745925512727859789049)
(719256274657338238, -7442465055852688040358540791)
(784232354887054504, -7293850484984624398670331929)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -41/35. 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 20:00:57 UTC · history