curve #609

y² = x³ − 1602690988179345137709737957237346859074207x + 764544453281342416894021418719617724754303455294486901846764994

a-invariants: [0, 0, 0, -1602690988179345137709737957237346859074207, 764544453281342416894021418719617724754303455294486901846764994]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 4118482257964122577897131179164327389821105780521383211377216820453903957341851856458623320
discriminant (Δ): 10952708829780421468861171587400899127893190865905526538295036012931357472584527738913978077083210441001377621186740755812000000
Faltings height: 23.1612
naive height: 303.1544
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 53, 61, 71, 83, 109, 3721035415760065191439636281120916611382462235483020039054732413643738793
regulator: 413197019706297631631648.459752336395905432378189929638606677701963822695691275002915
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:47:07 UTC
last updated: 2026-09-06 15:47:07 UTC

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

(-47896499018553275087, -29003411221458379583664270734250)
(901168480790596060045, 7217544363658596565088277989202)
(1300433786843564408038, -29657224678461064575561453783000)
(816009161664691129523, -301199042577598729779071105480)
(510972198443994952213, -8889598927933800976664248151850)
(-270140377532517652157, 34318832808899646548858354877840)
(592704011063221650778, 4778999506482507039353964978270)
(-1154959160405687586947, -32786455339347352592315462626530)
(278088366297032410213, -18448853111110616285496526362150)
(243692472071685164713, -19709202182111526736298095735650)
(641012011650018218213, -767988443213759077770183573850)
(21412250989782370908193, -3127879849036025464726991563027110)
(617343759516040036483, -3226623850070749184596567067040)
(815936663097305494963, -249203652010843332831917771400)
(614425040128862324365, 3430389291496536236484038179758)
(5643992108564476158313, 414133245477585792163297989155550)
(10084157165821124866507/9, 523171398516415218530235179744840/27)
(637935018080473072963, 1321543544595603340445213379600)
(3073831610791280349013, -157737307405731826263296453097450)
(153656056077434143418, 22845330477307110463675014476810)
(-110713739252055599657, 30669647670620164126792479447840)
(168377463622993671049, -22348625778406378759992816882750)
(10200812511437743276966/9, -542739122133898072752741415520620/27)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -829/254. 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 15:47:07 UTC · history