curve #3669

y² + xy + y = x³ + x² − 6769916094364055688x + 9632928478073408037976861656

a-invariants: [1, 1, 1, -6769916094364055688, 9632928478073408037976861656]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 7107840925923552917025
discriminant (Δ): -20228969844450156501638865914449837700655219218099132109375
Faltings height: 9.8904
naive height: 142.3931
primes of bad reduction: 3, 5, 7, 11, 19, 101, 239, 251, 311, 5519, 6229
regulator: 94303232.1646513097911848343000766350800731274740749780
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:30:08 UTC
last updated: 2026-09-29 19:30:08 UTC

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

(-3131394160, 11267160209367)
(-2600315810, 98258325017717)
(227934949831/169, 119438976869482964/2197)
(1371356051710/1849, 5633104990308653444/79507)
(4988400305335/1849, 8340682468870875644/79507)
(171133089049435/124609, 2379219832253811182434/43986977)
(14924346015677370/18241441, 5307944609064001358472372/77909194511)
(959611202914621660/506565049, 684665723564248482120273656/11401259557843)

Commentary

Specialization at t = 3/502 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 19:30:08 UTC · history