curve #3663

y² + xy + y = x³ − x² − 109108090361789456x + 16610051049747481430026755

a-invariants: [1, -1, 1, -109108090361789456, 16610051049747481430026755]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 1354000263461127126
discriminant (Δ): -36057448287999581220043877651515482965319003051051408
Faltings height: 8.8025
naive height: 129.6672
primes of bad reduction: 2, 3, 13, 19, 31, 37, 59, 109, 163, 241, 1051
regulator: 1000374.746634022565590571211471137541278824349154099
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:29:08 UTC
last updated: 2026-09-29 19:29:08 UTC

Witness: 7 independent points log in to add more points →

(-379900897, 1797752901801)
(-359499837, 3061647531345)
(-298303117, 4755452112795)
(70799951, 3039717512649)
(-651036216541/1681, 60694106738450307/68921)
(-99115571749/625, 85475371027991049/15625)
(441484866923/1681, 169794081523977945/68921)

Commentary

Specialization at t = 11/152 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), 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 19:29:08 UTC · history