curve #3700

y² + xy + y = x³ + x² − 495961009932247628605966051259x + 134400498515819422219587931004005638315321161

a-invariants: [1, 1, 1, -495961009932247628605966051259, 134400498515819422219587931004005638315321161]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 5069134872445093600035756292049418
discriminant (Δ): 4260941550781647062645903981724659183920255488442073528535354296869655297978368580911104
Faltings height: 15.7900
naive height: 216.7425
primes of bad reduction: 2, 3, 7, 17, 19, 23, 29, 41, 79, 199, 283, 523, 1699, 111953, 4410389
regulator: 329619816.1201763471072528551273309063459690320816330773312
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:45:49 UTC
last updated: 2026-09-30 01:45:49 UTC

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

(-188530524493389, 14872901774263563520294)
(394335412795827, 380565658717919009062)
(476791310038707, 2513870307317019541030)
(477465176597555, 2538809241104922829862)
(1007621279765939, 25645626114800238826790)
(1959282372887987, 81755374090987295700262)
(-13142978430763899/25, 1975783565185353296816534/125)
(-6465642611274077/81, 9601814670802461621681974/729)
(4239772442050008051/9409, 1417254992315346975483319270/912673)
(617894206805152235395/1495729, 226098082242386610002150848234/1829276567)

Commentary

Specialization at t = 2200/4811 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 2026-09-30 01:45:49 UTC · history