curve #3671

y² + xy + y = x³ − x² − 3961248372960718748x + 3046398107133845961154763606

a-invariants: [1, -1, 1, -3961248372960718748, 3046398107133845961154763606]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 18082853740749081030705
discriminant (Δ): -31089319686997066443492339123726659544632050995255376575
Faltings height: 9.4997
naive height: 140.0907
primes of bad reduction: 3, 5, 7, 19, 43, 89, 167, 227, 257, 1429, 8101
regulator: 13177245.0808072624433585074477024246288888506432028890
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:30:29 UTC
last updated: 2026-09-29 19:30:29 UTC

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

(-165337470, 60801572345107)
(-85442970840/121, 98633672055337862/1331)
(29706619261/16, 2927185674805233/64)
(101711508996/121, 23455434039076451/1331)
(314172219664/225, 52026095606770483/3375)
(434700373815/361, 32497214672774668/6859)
(2244894247440/2209, 870617521666124051/103823)
(5558658434715/2209, 9856146926316056546/103823)

Commentary

Specialization at t = -77/334 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 19:30:29 UTC · history