curve #2537

y² + xy + y = x³ − x² − 141773x − 20499919

a-invariants: [1, -1, 1, -141773, -20499919]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 6930
discriminant (Δ): 196932105562500
Faltings height: 1.6847
naive height: 47.1995
primes of bad reduction: 2, 3, 5, 7, 11
regulator: 6.066481053295083401928433407376978040253
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:07:47 UTC
last updated: 2026-09-26 20:07:47 UTC

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

(-595635/2704, 24984331/140608)

Commentary

Exact computational full 2- and 3-torsion family at (p,q)=(-9,2); 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 20:07:47 UTC · history