curve #3666

y² + xy + y = x³ − x² − 29136717497212840886x + 60533108273018762726766844101

a-invariants: [1, -1, 1, -29136717497212840886, 60533108273018762726766844101]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 830429816548607909946
discriminant (Δ): 117177695264410553328821428175374613454602585923761633024
Faltings height: 9.8379
naive height: 146.0692
primes of bad reduction: 2, 3, 13, 23, 29, 41, 53, 101, 883, 1613, 17021
regulator: 5410905.03338690483818208221308880725718759125399310593
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:29:38 UTC
last updated: 2026-09-29 19:29:38 UTC

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

(-638467179, 280848465250983)
(3140672159, 1804178911125)
(3166266237, 4590615769523)
(14248118079/4, 352574749380135/8)
(-179184831007/121, 421810834882290273/1331)
(30415594609/9, 695152370727065/27)
(109998783148293/37249, 112046486866613106851/7189057)
(183331065552791/58081, 50060627137782104445/13997521)

Commentary

Specialization at t = -859/928 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 19:29:38 UTC · history