curve #3629

y² + xy + y = x³ − 6166000806909576551x + 5882858220756238497524991998

a-invariants: [1, 0, 1, -6166000806909576551, 5882858220756238497524991998]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 145059433532653065217081920420450
discriminant (Δ): 52770149310761021642925563119248682216598567681372437500
Faltings height: 9.5787
naive height: 141.4103
primes of bad reduction: 2, 3, 5, 11, 19, 41, 53, 61, 103, 109, 263, 353, 443, 619, 3931, 31069
regulator: 2148808899929.7500086903095664327879794957006208454127468990355
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:22:49 UTC
last updated: 2026-09-29 19:22:49 UTC

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

(-11463097657/4, 33560084995653/8)
(-5284941717/4, 866193811757963/8)
(1047798917, 23926227272766)
(1234834062, 12318639748156)
(1383255792, 646457287141)
(4732653327, 287579480754286)
(23948824317/16, 169704240868009/64)
(13077417273/49, 22377249845745598/343)
(3023258161407/2209, 285063488160582662/103823)
(58763925544302/22801, 290312431099631267236/3442951)
(61438850069104508/40921609, 813878614377003895676838/261775532773)
(599507739526216983/64850809, 448915061488365584221885462/522243564877)

Commentary

Specialization at T = -30/41 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 19:22:49 UTC · history