curve #608

y² = x³ + x² − 973184317077048233239480365154220613923660x + 193615832649962368892393392627750252965319085982841345089284400

a-invariants: [0, 1, 0, -973184317077048233239480365154220613923660, 193615832649962368892393392627750252965319085982841345089284400]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 168979098173352083998670678858011914126690850746126480609555833436313402102894742081580070200120
discriminant (Δ): 42793795150908683102506355460476642413127991275720487254267323117993866206292557273288532148104447561517620299588169176244000000
Faltings height: 23.1871
naive height: 301.6578
primes of bad reduction: 2, 3, 5, 7, 13, 17, 41, 43, 53, 59, 179, 110202575387969, 1195744576663715752389508603292271452446924714141409557365802285619
regulator: 3384205942643454163485015.19850576365745472436349339841317486238899999002293113052124
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:46:55 UTC
last updated: 2026-09-06 15:46:55 UTC

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

(90219631473245950930, -10322298788902532447089278244950)
(895184614953403864980, -6308511281284622832712850605500)
(1895743109837635353640, 71845037836304835794802804488340)
(207966261653114574115, 469959647987182966982056108260)
(6257709797887859607550, 489028570446786597668491366737330)
(962495695193695032520, 12189510593678436268283911030140)
(-813711477240177562520, 21135923347329674422921344280500)
(24369785639421001361290, -3801231754937761698633220169718690)
(106866045831405438979195/121, 6385986495343602700502865922220760/1331)
(142048349573240755990, -7631698391377633491610707050490)
(4630505845085662192723, 308176348816585396008035653437504)
(-986946190856368966970, 13883401363356284018686769200950)
(-836808827920858407770, -20542894229128652136336143126250)
(116695508563355337989608/121, -16414015981679326764614242538972676/1331)
(-856995328062620212940, -19955393921741989994846518393620)
(-1033205059578704649110, -9805879622678688781418996401410)
(-10086010207773081274295/16, -1509902261738162320233357874828275/64)
(456178663117376040018445/484, -113514140382838476132511446785782875/10648)
(865820940632720697970, 267565283176408822182663449910)
(4311839800617282279730, -275976384305305322042458334426250)
(252763330649445847265140/1369, -227811444673962289353080314021798860/50653)
(-467395662018190082435/4, -139882453113122078655538834094685/8)
(235974390210510556165/4, -93435582378547845312446850378735/8)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -67/142. 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 15:46:55 UTC · history