curve #3648

y² = x³ + x² − 1248869357683141748660243160x − 4876649751055622525002422594205126924716

a-invariants: [0, 1, 0, -1248869357683141748660243160, -4876649751055622525002422594205126924716]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 66489049159927911945099447956450941788165109722672
discriminant (Δ): 114387414089721574790994322680274155266079235884247851872967832301241258376896909312
Faltings height: 14.6288
naive height: 198.7897
primes of bad reduction: 2, 3, 11, 43, 59, 67, 73, 347, 787, 907, 1097, 2113, 5659, 21517, 10645571, 39297109
regulator: 12412053435347711.537700615818640055311111892523115259632044286084318
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:26:36 UTC
last updated: 2026-09-29 19:26:36 UTC

Witness: 15 independent points log in to add more points →

(-16003029846420, 104931961248755966622)
(-11138017107933, 87473122005189162396)
(-7808317302402, 66323802436362562080)
(1887399160320750, 81982156390805741994528)
(121743491764184454, 42478445202717796605052632)
(-25879752251231239428/2301289, 307245001477945283510248993590/3491055413)
(-4811400031279389/625, 1022360636501127748203984/15625)
(-3553826403626106/289, 456173382553645720202712/4913)
(-181898675677194/25, 7730694939400586495496/125)
(336437066233732/9, 701967728028195414170/27)
(1933578120393684/49, 29282765019897822775518/343)
(93335319290666406/2401, 8580428512188730511408424/117649)
(598011854981587356/361, 462344408670181128858396234/6859)
(9407382976931797398/121801, 25484006268034431464358162360/42508549)
(21659978818433618046/494209, 54464269435368534247761774816/347428927)

Commentary

Specialization at T = 544 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 19:26:36 UTC · history