curve #3672

y² + xy + y = x³ − 165870106386207383082x + 797040010380379297766763083200

a-invariants: [1, 0, 1, -165870106386207383082, 797040010380379297766763083200]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 19479441722836754745294
discriminant (Δ): 17630406549254631700419091666575701472304933193575111002530124
Faltings height: 10.5232
naive height: 151.2868
primes of bad reduction: 2, 3, 13, 17, 29, 83, 179, 373, 863, 5099, 20773
regulator: 2621682.20212454681331888083268063658340443360766317617
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:30:39 UTC
last updated: 2026-09-29 19:30:39 UTC

Witness: 8 independent points log in to add more points →

(-14271692950, 507364417949724)
(5863018595, 161495305451919)
(-23941054950493/2209, 119360523316926264579/103823)
(-861979603571/144, 2168934409983615277/1728)
(300798899906/49, 34536394568359188/343)
(530282983349/121, 522803097532696821/1331)
(584062752839/121, 439140330226730871/1331)
(1070405325815/121, 197368981706623959/1331)

Commentary

Specialization at t = 199/664 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 19:30:39 UTC · history