curve #3619

y² + xy = x³ − 306245784794605070x + 64040074904742348830845956

a-invariants: [1, 0, 0, -306245784794605070, 64040074904742348830845956]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 58287634249314930726735678
discriminant (Δ): 66501043597772984691535672771404066586585155058532352
Faltings height: 8.9255
naive height: 132.4031
primes of bad reduction: 2, 3, 11, 19, 31, 67, 83, 101, 107, 163, 199, 211, 283, 1171
regulator: 1968061523.313764057184592244702821463028910678164726865356
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:20:47 UTC
last updated: 2026-09-29 19:20:47 UTC

Witness: 10 independent points log in to add more points →

(-398130470, 11084378779744)
(282934750, 204830623366)
(383433340, 1728400859746)
(447621052, 4079657364514)
(1236546652, 39699401800354)
(-13851337964/25, 997185912343394/125)
(-7981736/9, 216525318370006/27)
(10924815979/100, 5646818916905563/1000)
(260337695122/529, 69378772748445848/12167)
(59691365402140/157609, 96504804645491159242/62570773)

Commentary

Specialization at T = -17/2 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 19:20:47 UTC · history