curve #603

y² + xy = x³ − 47750477658308337922522338108829117329485x + 4054931615306809915439526124593132171571389788807470809162897

a-invariants: [1, 0, 0, -47750477658308337922522338108829117329485, 4054931615306809915439526124593132171571389788807470809162897]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 16854864852052069315413395831973557760017593822284352366748443540111622952880842175074981539057190
discriminant (Δ): -135067107355235827701563800268694131505662911123744756908069759287770973858669928043476649844893154734718961519958224000000
Faltings height: 22.2491
naive height: 292.6332
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 37, 127, 255586897, 27490142467359115890200291762103480608547729694090727479231177319757972234785423
regulator: 12815207646561990534272173.4343510057661636705941743969084899338763872316991961313001
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:45:56 UTC
last updated: 2026-09-06 15:45:56 UTC

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

(-162526653026313769466, 2742723015400181888064300656983)
(-221723272868789275766, -1934461031511398477835560411417)
(-237552322989938010866, -1411675653213953695487775755417)
(140432317759450639342, 344558015532328391598583143463)
(35492754875596525234, 1550756945855376126762109161583)
(477095425400070078511/4, -1899356292909840173552585043511/8)
(427552486847326277164, 7861054219850392425558786629173)
(57049533531684711514, 1231448481232730350269309641263)
(24178749183854672134, -1707196525720179415700722997417)
(-251891439809450586866, -317069857871225830433502731417)
(12718711336175444953756/9, -1418172400476652156241754068832859/27)
(-185276528484245750286, 2557716623625385910701182971823)
(942455482795013541634, 28216347098937058203381043359583)
(-31464935315140346006, -2350796821010176486218266631857)
(89688439587094964434, -702652862551293597534739947617)
(131856941322002028842866/1849, -79997681198337701683568094821821619/79507)
(787042888277162975614, -21307190244849138609941823277637)
(98912185880768316784, -547308673751701909846397001767)
(256228652228078887084, -2939745244760404707916535499617)
(2088467978084906122432, 94939966430229646848715174928983)
(3102426401727248288465812/37249, -5821035371158921514161947244835612753/7189057)
(919950260704822908106/9, -13282832619262076874888016573259/27)
(-150559441996951175006/25, 260476864386442084702512884534387/125)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -831/416. 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 15:45:57 UTC · history