curve #738

y² + xy = x³ − 2235623135069800412509513626575x + 1269469751113808831812336410678494175567490041

a-invariants: [1, 0, 0, -2235623135069800412509513626575, 1269469751113808831812336410678494175567490041]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 1634992090157394062979046231203825710996833965246029548763487852386502834
naive height: 221.2598
Faltings height: 16.3161
discriminant (Δ): 18923699053448771374187322251127587284235563484954725118277556743145170050467668724467604992
primes of bad reduction: 2, 3, 13, 19, 29, 31, 37, 61, 167, 403079, 3659501, 15929437806119, 2234508752346727, 62010492272670768451
regulator: 91446645825001322758.185080154547684559780342684784416373177717971050253840924635
submitted by: Valery Asiryan
submitted at: 2026-09-16 17:41:35 UTC
last updated: 2026-09-16 17:41:35 UTC

Witness: 21 independent points log in to add more points →

(-1721109311031716, -4349922869857991908595)
(-31465663064561828039/18496, -30726385035563916803665631129/2515456)
(-1693275654637376, -14144459243242648837109)
(-25167999406043531/16, -1913573855193503488201571/64)
(-1454369273941514, -38008209230150733884579)
(-278613934726796663/256, -201224611915679431969914371/4096)
(-979256578671998, -50196281590203492836471)
(595997557299994, -12196309433189880537275)
(912358458254303782/1369, -440284626761681226265044791/50653)
(733670983668502, -4916634404644238608931)
(6738059999057074/9, -105812918225215923264233/27)
(944000954905564, -523750081740825521129)
(944228543449924, -611634680233563155009)
(48096793073341296/49, -1563194409272788564711965/343)
(1000840570247800, -5872925560704301260221)
(1098792438831838, -11815469964252406992587)
(1401516201180880, -29818327770617315444069)
(2190620068284502, -82972599590894280308771)
(2788263238596256, -129279043842795015540107)
(79011716118192613/16, -21289391738541849181771529/64)
(12168411187847374, -1332608328766037143423859)

Commentary

Specialization T=-13/47 of the family identified by wgxli in the commentary to curve #302. The points were obtained by specializing reconstructed generic sections and running minimized quartic searches. The 21 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 17:41:35 UTC · history