curve #1172

y² + xy + y = x³ + x² − 2365x + 43251

a-invariants: [1, 1, 1, -2365, 43251]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/5ℤ
conductor (N): 5302 ☆ record for ℤ/5ℤ torsion, rank ≥ 2
discriminant (Δ): 1242025312 ☆ record for ℤ/5ℤ torsion, rank ≥ 2
Faltings height: 0.6722 ☆ record for ℤ/5ℤ torsion, rank ≥ 2
naive height: 34.9192
primes of bad reduction: 2, 11, 241
regulator: 3.17968053395106900607241719756962463160199
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:17 UTC
last updated: 2026-09-23 21:38:17 UTC

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

(27, -12)
(35, 52)

Commentary

Cremona 5302i1, rank exactly 2 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.00000-09999 . 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 21:38:17 UTC · history