curve #3715

y² + xy + y = x³ − 2684837227781839213x + 1654025607387102256872495156

a-invariants: [1, 0, 1, -2684837227781839213, 1654025607387102256872495156]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 169544208072571930
discriminant (Δ): 56742026048583851334135583943924588786582150899414062500
Faltings height: 9.4792
naive height: 138.9161
primes of bad reduction: 2, 5, 13, 17, 31, 73, 1609, 1699, 12401
regulator: 781771.35761373214987212794781968623651621329718360
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:48:37 UTC
last updated: 2026-09-30 01:48:37 UTC

Witness: 6 independent points log in to add more points →

(13954857175, 1637591576321412)
(-2801199722691425/2053489, 155101909598953581456144/2942649737)
(-83834917710200/137641, 2826330296615650158132/51064811)
(9815257612300/841, 30461968473530239143/24389)
(26282754809300/259081, 4903625179966746304848/131872229)
(7816008668421825/9467929, 13461014964193448416596/29132817533)

Commentary

Specialization at t = -125551/56575 of the universal curve with a 6-torsion point, y^2 + txy + (t+2)y = x^3 (as searched in arXiv:2003.00077, Sec. 13), 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 01:48:37 UTC · history