curve #2536

y² + xy = x³ − x² − 57699x − 647595

a-invariants: [1, -1, 0, -57699, -647595]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 6930
discriminant (Δ): 12104550027662400
Faltings height: 1.7758
naive height: 44.5026
primes of bad reduction: 2, 3, 5, 7, 11
regulator: 7.651180632372028931438602710891376491612
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:07:43 UTC
last updated: 2026-09-26 20:07:43 UTC

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

(-585051/2704, 197051061/140608)

Commentary

Exact computational full 2- and 3-torsion family at (p,q)=(-21,1); 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=6930; local exponents {'2': 1, '3': 2, '5': 1, '7': 1, '11': 1}.

last edited by Natanael Wildner Fraga at 2026-09-26 20:07:43 UTC · history