curve #610

y² + xy = x³ − 2217570089183323351975962968677939260666154425x + 40214866532442415571867494413978840703302496270716933910245865205625

a-invariants: [1, 0, 0, -2217570089183323351975962968677939260666154425, 40214866532442415571867494413978840703302496270716933910245865205625]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 485584274111793367302945692155141768368238327423581460030101770225568239962747898276490356665639290
discriminant (Δ): -715453888397580116007427890029949801368222931532507493863860632004113500168914116667588321360136397703659182422231647503629551616000000
Faltings height: 24.8206
naive height: 324.8529
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 61, 401, 1301, 15361636027, 21932791490552348952479726763618916283981682294069713337742694520977687
regulator: 6835054646037091756050758.24265023662183816219322436016622526176423970401021434858650
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:47:19 UTC
last updated: 2026-09-06 15:47:19 UTC

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

(62896166784229047842850, -12229100653817916767794015595453925)
(24868931680269371951610, 668431413867252783127408642505715)
(43666202380864674776750, -5161586579042863652004623159991925)
(-54376104885371841306150, 143373156839757699628916131857075)
(-1458027316755551304750, -6591285516586270918669456063991925)
(66392326170578393050650, 13624928798838895893446565319713075)
(30446324323985104150200, 959740307136088072154059270638075)
(109497294239221251829050, -33320143924436031667910105574506925)
(194928663309115592191920, -83753729092587566827770348046506645)
(26917005503104669217850, -162998886509450069463717878678925)
(15377460574020470734430778/529, -6837365641868109885158556952494776091/12167)
(27138544613917759979970, 144208966465229929055143318314555)
(-47341981997860517320230, 6252454309373937883216638069592755)
(-50142621737848268953110, 5033579999646327292936972034862675)
(27713277180477421999050, 207947057420677120996456882743075)
(7930013615047411875663650/361, 9943933831234776656925491051508284225/6859)
(244847942788905983863750/9, 3877185848427760796552441835440725/27)
(-53849360373474179236710, 1865400798084160685523935534602035)
(58732310001476121996570, 10609808208528354127127645511959475)
(25830643358691221899290, -410343125035855767426042556213965)
(27400631357410632468732, 155859025855657305308993308462227)
(28294146388632411374100, 348911757440847524485441076814825)
(23892767628510851756270, 933010628513785453652755667699975)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 2004/247. 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 15:47:19 UTC · history