curve #3665

y² + xy + y = x³ + x² − 210050813820678350x + 229959419870628137482189067

a-invariants: [1, 1, 1, -210050813820678350, 229959419870628137482189067]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 1906431864891437910
discriminant (Δ): -22251602290576149169521136749613131550573787537909760000
Faltings height: 9.3010
naive height: 134.9230
primes of bad reduction: 2, 3, 5, 13, 37, 53, 89, 157, 173, 317, 3253
regulator: 1175224.654861029311954531662723823714841104790808551
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:29:28 UTC
last updated: 2026-09-29 19:29:28 UTC

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

(-706779663, 5035839823831)
(-663593673, 8782699502221)
(122324777, 14355961598551)
(-4401233669378111/26656569, 2219789865056601934420853/137627865747)
(-38177632703/81, 10917085684656799/729)
(-6476179607/9, 78813723800477/27)
(152929236103/324, 89586046963127087/5832)

Commentary

Specialization at t = -13/80 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), 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 19:29:28 UTC · history