curve #1171

y² + y = x³ + x² − 840x + 39800

a-invariants: [0, 1, 1, -840, 39800]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/5ℤ
conductor (N): 13881
discriminant (Δ): -656000554923
Faltings height: 0.9497
naive height: 34.7204 ☆ record for ℤ/5ℤ torsion, rank ≥ 2
primes of bad reduction: 3, 7, 661
regulator: 2.21353449593632858694046066950438543037200
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:16 UTC
last updated: 2026-09-23 21:38:16 UTC

Witness: 2 independent points log in to add more points →

(0, 199)
(-42, 52)

Commentary

Cremona 13881d1, rank exactly 2 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.10000-19999 . Found by screening the local allgens tables for missing torsion size records; torsion and independence independently verified by the site.

last edited by David Renshaw at 2026-09-23 21:38:16 UTC · history