curve #463

y² + xy = x³ − 4682646150937436689802827639391590x + 118660653877767113794215602966189789176568569086692

a-invariants: [1, 0, 0, -4682646150937436689802827639391590, 118660653877767113794215602966189789176568569086692]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 246487897243725896044435022343614140294003013883226125048980054731003325130
discriminant (Δ): 488629359416104574728554950607882050032668228093550025226917655718646710468793242267356967557120000000
Faltings height: 18.2762
naive height: 244.2011
primes of bad reduction: 2, 3, 5, 7, 13, 19, 31, 47, 193, 1489657069, 12380920864241, 130895646239455408856934499736976436958033
regulator: 40771069034647434994.0276133238166140394084724288500817109028367864395589160546
submitted by: Bhavik Mehta
submitted at: 2026-08-30 20:00:45 UTC
last updated: 2026-08-30 20:00:45 UTC

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

(164176299210670604, 61441562939603718870044618)
(-78680401433729056, -117479567535362929788502)
(425632816878281324, 274289292805479560762283338)
(19527401906378444, -5887861945466994790849942)
(113367005608299524, 32323438430883394379026538)
(17247802132172924, 6559442877571183195526138)
(-65737110804207076, 11933568359799387106736138)
(31341011486411582, -1639139334684609426874060)
(-78451812269454676, 1782243970464564934275338)
(-2216076320742076, 11358999604646376971160938)
(49840643780576324, -3013832897969847239519662)
(31720873439863724, -1428638856638953942326262)
(29767068272314412, -2376543945011638064282230)
(27884629482322652, -3125470544724181274454262)
(-2661406785133876, -11450075506844375781267862)
(820750715440524908, 741052700466077860840215626)
(-74937472084620676, -6981804759340349178092662)
(76596614912092652, 14470014540091465385105738)
(-1861606671197716, -11285895960200648161424182)
(127727738994959324, -40054402066749323240402662)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -39/16. 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 20:00:45 UTC · history