curve #3633

y² + xy + y = x³ + x² − 1621798311910830552290x + 24644776741169831070627868749551

a-invariants: [1, 1, 1, -1621798311910830552290, 24644776741169831070627868749551]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 5823102912675542002999614449352786
discriminant (Δ): 10623251366845607892971993300969516309003874094605629097183347712
Faltings height: 11.0726
naive height: 158.1271
primes of bad reduction: 2, 3, 11, 19, 23, 47, 97, 107, 109, 271, 353, 563, 1153, 1291, 5051, 9377
regulator: 210733412349.85980905463824595806087277400414006391114694990376
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:23:33 UTC
last updated: 2026-09-29 19:23:33 UTC

Witness: 12 independent points log in to add more points →

(-45953609519, 1459630107017771)
(-43086734727, 3812328161487427)
(11190585661, 2810206492805987)
(11243676297, 2798427858983203)
(15128525969, 1889928775149059)
(26571554049, 558425733441979)
(26970535049, 722924255973971)
(130623884789, 45183805517420039)
(365819895657, 219970069911133219)
(259430756265/289, 23658853853056453091/4913)
(30581525469447/1849, 121764821153031601813/79507)
(454463771709241/29929, 9713449321425190752607/5177717)

Commentary

Specialization at T = -89/9 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 19:23:33 UTC · history