curve #3621

y² + xy = x³ + x² − 79670263216654392x + 8794677070351063157969340

a-invariants: [1, 1, 0, -79670263216654392, 8794677070351063157969340]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 368839630407777086661212802
discriminant (Δ): -1049134010807405039233058690603118803218714473891468
Faltings height: 8.5849
naive height: 128.3955
primes of bad reduction: 2, 3, 7, 11, 19, 23, 29, 97, 139, 227, 263, 463, 10789, 15667
regulator: 1848320349.069389872689525778733370196299938950403847390258
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:21:09 UTC
last updated: 2026-09-29 19:21:09 UTC

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

(133542924, 732629145414)
(151273098, 452002998690)
(163291434, 373044958914)
(3591693158, 214605795358582)
(8806374839, 825986676840964)
(-1217692388083/6889, 2382042771612811050/571787)
(-1150772978834/6889, 2388035041786164668/571787)
(-29445706380/121, 4939436292474150/1331)
(55797957286/289, 3835765425557732/4913)
(91450124226902/450241, 300877330951492466172/302111711)

Commentary

Specialization at T = 26/9 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 19:21:09 UTC · history