curve #606

y² + xy = x³ − 526772798870782915067032030614494198900560x + 142401047592558816837792412780017813107066634367624720197155072

a-invariants: [1, 0, 0, -526772798870782915067032030614494198900560, 142401047592558816837792412780017813107066634367624720197155072]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 359358360841115150113680865855445183755920843390259374364408065852737237862742266569139990
discriminant (Δ): 595012459101326259334700906472707525020750931016512030508335461016388987009434895429598496845521902710257327572189767680000000
Faltings height: 22.9032
naive height: 299.8164
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 37, 53, 59, 73, 6761, 1840051, 2069282471130041, 1227450710329226656621849, 52689704047356459006530071
regulator: 292078245197911549608958.912386158976161237405547228197059417441924231821140717056945
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:46:32 UTC
last updated: 2026-09-06 15:46:32 UTC

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

(353140197944890104224, 644829219371487599622746122928)
(69324238868102433152, -10306118869635721496574021174832)
(-66057905656041579616, 13300763213748239584113968485808)
(785076981292858692030656/2209, -28693243095086540837167666821001616/103823)
(173593655885855812544, -7495854475420025067657599658832)
(354353672354539446134, 481799869220473754116399149308)
(-3244252981019100070909/4, 48073656441890560098425214572599/8)
(-834998683950385757866, -274807449012186916037270031442)
(1199435147283613887584, 35158672083110995714035936071408)
(934249404468737342384, 21580019942029626330693778075808)
(2482264432319507628884, -118868092103414998721486857240192)
(573259100181708358784, 5367642838169608314722782912208)
(42398897400972874386464/121, -1220880146071320179598428877629552/1331)
(-15148766852646731525859904/18769, 16621104467051363202772166625136651024/2571353)
(2171733202998387002912, -96131202374947683921567831332560)
(502702667039964437984, -2151495598936594058645562989392)
(242695410623279060384, -5371287031700636535354808666192)
(-786550058216847818866, 8374149646809868136323707989558)
(793666993559835842384, 14975162582701793145421052581808)
(-687529353827695239616, -13400746738239771636189727294192)
(577007800112001847064, 5527825748870914398440419799288)
(-150323463802589230216, 14771272065551003581423140973208)
(9785434473805675779269/16, 447965316792815625041142994526087/64)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -33/113. 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 15:46:32 UTC · history