curve #3627

y² + xy = x³ − 5572586182117973719x + 443264061932761831373441736809

a-invariants: [1, 0, 0, -5572586182117973719, 443264061932761831373441736809]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 2355694911715481929719037783146
discriminant (Δ): -84869593187040709081893813621983900101006839179127142351896576
Faltings height: 10.5620
naive height: 150.0511
primes of bad reduction: 2, 3, 11, 31, 37, 41, 127, 271, 293, 397, 577, 823, 1483, 24473
regulator: 26818075718.6130891519428466423179148041308382365198948995346
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:22:16 UTC
last updated: 2026-09-29 19:22:16 UTC

Witness: 11 independent points log in to add more points →

(-7844380666, 65418087507317)
(61147647887/4, 15860269155571273/8)
(30679895894, 5399052557203157)
(60656141270, 14942170130125013)
(-98733789549706/14161, 637662179867025215923/1685159)
(-882108559786/2401, 78504775673157189413/117649)
(-385088931610/361, 4590897181431263783/6859)
(-180115082194/25, 41354102929278889/125)
(22601649254/25, 82817985896384257/125)
(67634849354/49, 227054884423953851/343)
(39273366634952/17161, 1495437380966100961999/2248091)

Commentary

Specialization at T = 115/2 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 19:22:16 UTC · history