curve #452

y² + xy = x³ − 55000206503542170766038007004896613x + 5296677150314724213014972255851724752418661641474817

a-invariants: [1, 0, 0, -55000206503542170766038007004896613, 5296677150314724213014972255851724752418661641474817]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 2370458085180475003482735023025170108836693446674620297777501668599438211647550
discriminant (Δ): -1471548838868147549133553708559492561674695419611207333816195379402867092381604973017484205937413201920000
Faltings height: 18.9265
naive height: 251.7210
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 67, 48699141226693, 313741437670571, 35290476238827690372768417782626907968141
regulator: 160280604755817816591.185079863063327786954357799276134982025293230315618192913
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:04 UTC
last updated: 2026-08-30 15:55:04 UTC

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

(163755056272540722, 26102197601152179420807399)
(-197572972942892558, -91929074348041884862094681)
(176498478865401682, -32976572546372706447703001)
(2139194728885146922, -3110775664649129924047307201)
(324756012103404442, 147261328040051441332421239)
(295670061256904862, 121993735826229706786585359)
(223672653539085922, 64690464270159869489758279)
(118987737166170862, 20903710771345285947917359)
(-267215363843855438, -30221150767928227239959321)
(503798331385531567/4, 153341402070661451648313785/8)
(117378210822725842, -21402022445571714147667961)
(445474418786112562, 263056567923495052498536679)
(97732045716674722, 29238454850225985847285399)
(95076809147933602, -30444856805500685223942761)
(264380813210707378, -96099741082873399692211865)
(-200046647835341318, 91069671482131012252395319)
(591674301971852242, -424130768361456984296825081)
(141827552473968363442/961, 591373498823731363564773250489/29791)
(659550891840133282, -505895984330369793635461481)
(-155587825844828172686/21025, -230232370401279827009574053217353/3048625)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -13/19. 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 15:55:04 UTC · history