curve #3638

y² + xy + y = x³ − 26584447235992210259588x + 1883021637908148057017778972149102

a-invariants: [1, 0, 1, -26584447235992210259588, 1883021637908148057017778972149102]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 551664926630833203374698683114972985506
discriminant (Δ): -329334338676931735094752960766191284170572391342767104152001882159708
Faltings height: 11.8756
naive height: 166.7595
primes of bad reduction: 2, 3, 11, 23, 47, 53, 59, 89, 157, 251, 317, 521, 7607, 9533, 55997, 95569
regulator: 6963504929908.204124646260172139829621030678306948989635030177765
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:24:34 UTC
last updated: 2026-09-29 19:24:34 UTC

Witness: 13 independent points log in to add more points →

(-183830241354, 23617228343695558)
(-164268709683, 42630508966552441)
(-27740385429, 50981750012106133)
(601674726, 43209103667138278)
(23241539760, 35745093131163493)
(25080144429, 35100637236226177)
(62530420215, 21568053176674081)
(-124140812528490/5041, 17976836370414805133573/357911)
(838667783619/49, 12984370708861759339/343)
(214961515825449/256, 3096789656852165404867/4096)
(1486146486122229/784, 57087395558460140154721/21952)
(519209165870076/961, 11353025978269864997758/29791)
(3043703389005759/44521, 184343767087289987445037/9393931)

Commentary

Specialization at T = 54/47 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:24:34 UTC · history