curve #600

y² + xy = x³ − 267363133273234282656999992433505375455x + 1676555666364951857445303859179396712885703435256389576025

a-invariants: [1, 0, 0, -267363133273234282656999992433505375455, 1676555666364951857445303859179396712885703435256389576025]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 228001852874781479760816970618209987557663751437348824206835929566733219877581322524771870
discriminant (Δ): 8881171505845574880320624088491350678064646227084932972708101647246459807244607224779794011557733857450397696000000
Faltings height: 20.9111
naive height: 277.0586
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 73, 1709, 3314683491850541, 17410932364273432099, 4771574992486411657388928663090308317459423
regulator: 852515066835138080736132.952866188449696683482796215903373554294354789551981497515062
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:45:21 UTC
last updated: 2026-09-06 15:45:21 UTC

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

(5274144479892974310, 20326156549867454601635704845)
(-18063643653731137110, -24739187830301371417800989355)
(16007500505497530150, 38710505857333175175918296205)
(12722882135809060710, -18286633763407101533044125555)
(24503850479942519310, 99187646730311960646952510845)
(-15015532175124588690, 48017291875482051281736421845)
(10364605141976808360, -4342865398751400163825018155)
(4745750367116116710, 22684828930721846721950738445)
(4685453327651317560, -22949948944855114696287639405)
(10003371778091249910, 1742044486325572801280329245)
(-13783374406172250, -40990740782695233939348697395)
(30703415650923251830/9, 765654823124791742937868000895/27)
(4473664910399593260, 23874611753709251425594975095)
(20921560930903431150, 72391393339869252977857910205)
(10570006271552686310, 5608918424495036705490456845)
(37690208379725880330, -212462884087850060697895862175)
(15280012238821485207030/1369, -461095811273208270915286542147015/50653)
(18052647267607397366310/961, -1706984017961632282770721843282605/29791)
(14918185821122049510, 31749942579900791581665381645)
(16270794333695552310, -40420970275931139941445301155)
(8310887794442798280, 5345165931146410845471955875)
(10055506774400062470, 2197385238335067264272106285)
(12936914509935444435/4, -232643653036143524536290487965/8)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 28/117. 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 15:45:21 UTC · history