curve #601

y² + xy + y = x³ − x² − 4326804635479538494575503235125633870567x + 114325279417711481909403784860112770146861553949957593778591

a-invariants: [1, -1, 1, -4326804635479538494575503235125633870567, 114325279417711481909403784860112770146861553949957593778591]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 530500502593057961420036692544358590035411455325690626851040620565705409063433583480522570
discriminant (Δ): -462155413041608288091056862837014184008286992858321568755049644026481538233970319862396301064751594597674435872000000000
Faltings height: 21.7215
naive height: 285.4959
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 41, 71, 101, 39119573617, 5483272733966728699, 41296297010177354778585591507548109010253977861
regulator: 476678640711692755093848.460724425337555568201413223228202959764222945229417176233270
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:45:33 UTC
last updated: 2026-09-06 15:45:33 UTC

Witness: 23 independent points log in to add more points →

(40158774589305851971, -73015440340778830330179983986)
(-74039817233057181469, 169713388732479893437279656494)
(69546330869914832971, -387022072192695900987784724986)
(171087061558117220371, -2093302935284804190684050517586)
(1868752084528448611, -325954107746241634349535365266)
(-34483346064982554941, -471724277877709464751167737074)
(32149138604578460671, 91926763498395889475392583714)
(32832394079764138351, -87510755655075361816308060706)
(36265969014519723971, -71465115888076082273633771986)
(29522136131294786471, 110990874551246846267766915514)
(293505988914939530221, -4912095307492142091441094965736)
(-22787008418788892269, 448428489473346330736975986094)
(78724801810516915776379/1849, -6786797255865748893736648700378902/79507)
(76944175596218063099/4, -1563841425376743331786775160323/8)
(150209267139468452771, 1689243728306393231431376714414)
(30308906801398504771, 105010603834626209523013370414)
(114852513191812990631, -1064148417893776715862025851206)
(466577463963585854371, 9983330977705618342469045546414)
(216270814446795873971, 3048641109947657742773421498014)
(38558790626936630971, -69407326903013504879133899986)
(2075251557014565421, -324584363121989869084376580136)
(-64904344906981900829, 348911657581124836219995333614)
(-60330172293948077029, -394685481482170020747820454986)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -172/17. The 6 points beyond the seventeen generic sections come from the surface's rational quadratic sections: writing x_tau = a/c^2 and y_tau = b/c^3 for a trace of height 10, the slope of the bisection is forced to be kappa/c with kappa = -b*a^(-1) mod c^2 of degree 5, so each section's point on this fibre is obtained by one polynomial evaluation and one square test rather than by any search. Claimed lower bound: rank >= 23.

last edited by Bhavik Mehta at 2026-09-06 15:45:33 UTC · history