curve #599

y² + xy = x³ − 37911831448895638203621690173938156355x + 90691670784852986494767987482904300661683284395472624577

a-invariants: [1, 0, 0, -37911831448895638203621690173938156355, 90691670784852986494767987482904300661683284395472624577]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 3120065552430437518084446980377456978556342298121691656956566322835163731063945882578030
discriminant (Δ): -65770874146009551411416078978771259752043406805865231058980612946969450984819909568084282222646179399040000000000
Faltings height: 20.4633
naive height: 271.2173
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 43, 31183, 133391, 245893367, 2187124087688004925459, 9076706842899395697742478234233539422417
regulator: 616915036962952511384274.429113275492312362264657863368570567913264444125356891741200
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:45:09 UTC
last updated: 2026-09-06 15:45:09 UTC

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

(10400118073602942094, 28658444828521471144532909353)
(-6869885821589511506, -5187975271997018792723968247)
(2894917750466992594, 2280578082790935510887889853)
(3183090652798756494, 1505376821122742029339527753)
(95985094197327826714, 938496493984386818556656719813)
(3590994773247056494, -925869364932599754190384247)
(3940466906798198494, -1576732187336854975969738247)
(48264129288729776206/49, -2527628659142914535901695764721/343)
(-5508845383664178146, 11504915396418176941803032233)
(2302571996016992494, -3950292725917596552320176247)
(52768254457124576494, 380818536312754104426380047753)
(956163863257818992494, 29565791302532754884146551823753)
(115763893806754277494, 1243819167717227784291985333753)
(33745239511072989586/9, -30240234045885699076966804609/27)
(-7089700103115237506, -1766233543922015402968006247)
(5724430955224457494, -7826398635778546778660146247)
(3817111376531750434, -1262772906523555916972778227)
(7090807986404072494, -13356228673284921200723296247)
(3938773618232029294, -1572080613968612204173897847)
(1713103296272144494, 5547275983084318954010959753)
(2119983298938512494, 4455011121153293955347743753)
(1397617332139828924, 6358887622306797601937304043)
(-5679359777621907506, 11082334042150727167354423753)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 275/13. 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 15:45:09 UTC · history