curve #459

y² + xy = x³ − 1928281706386442181413475191954491925x + 1041272774542851005444015987964798785676127762575830625

a-invariants: [1, 0, 0, -1928281706386442181413475191954491925, 1041272774542851005444015987964798785676127762575830625]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 306630491920777660800789905311462621887091488323154470032105508244122033478180321990
discriminant (Δ): -9523712949064575272159815189891085656872405169059420712094224530887379514870038402491873816813367453696000000
Faltings height: 19.7229
naive height: 262.2832
primes of bad reduction: 2, 3, 5, 7, 13, 17, 37, 53, 853, 723071, 1031813399, 4250256977, 33664961833630290267498865544386230765212873423
regulator: 770691786839733960707.601050797337180166880488833030170375156158952417938886978
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:57:08 UTC
last updated: 2026-08-30 15:57:08 UTC

Witness: 20 independent points log in to add more points →

(776096304873893650, -110467382469190686452369225)
(3249383160353677630, 5392970355484074740412071395)
(-1493698116763758350, -767400011165300134180524425)
(383822445707741050, 598079902462208890594978375)
(263425887055935250, 742693355349190190997995575)
(742054005217641490, 137808945671122395835505335)
(8003725435241809090/9, -4657794000112840877711462995/27)
(502437414977501650, 446395436373181106434475575)
(-5049530415477656705/4, -9678865113552357282334340515/8)
(802096136354440210, 103153303705612373479522615)
(469837175643031570, 488885955225081533297521975)
(701802284745361810, -183456951927329020143910985)
(949766844277928050, -258067102581468619502683625)
(1279452454398048850, -817673448591136098129588425)
(-10134597024761813150/9, -36070678071480945395413159475/27)
(825899050877540650, 109818573493868614431108775)
(1492769253713359450, -1220336221580496334593699425)
(10353218732586035100850/961, -1045102686364836180490864570393175/29791)
(983770841715007720, 310457507940596917359802975)
(1843761613903631250, -1937464607926962836124004425)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -45/23. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 15:57:08 UTC · history