curve #2523

y² + xy + y = x³ − x² − 12028568x − 16053857269

a-invariants: [1, -1, 1, -12028568, -16053857269]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 18810
discriminant (Δ): 4230728595002250000
Faltings height: 2.6897
naive height: 60.5220
primes of bad reduction: 2, 3, 5, 11, 19
regulator: 9.552825382243724226315623904860173623623
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:06:54 UTC
last updated: 2026-09-26 20:06:54 UTC

Witness: 1 independent point log in to add more points →

(-488051505/243049, 157187315029/119823157)

Commentary

Exact computational full 2- and 3-torsion family at (p,q)=(-15,4); Velu quotient by rational order-3 subgroup. The mapped rank >=1 witness is non-torsion by the original exact 2-descent Kummer certificate and finite isogeny kernel. Conductor N=18810; local exponents {'2': 1, '3': 2, '5': 1, '11': 1, '19': 1}.

last edited by Natanael Wildner Fraga at 2026-09-26 20:06:54 UTC · history