curve #43

y² + xy = x³ + x² − 2582x + 48720

a-invariants: [1, 1, 0, -2582, 48720]
c-invariants: [123961, -43023725]
rank (lower bound): ≥ 6
torsion subgroup: trivial
conductor (N): 5187563742 ★ record for rank ≥ 6
discriminant (Δ): 31125382452
Faltings height: 0.8126
naive height: 35.1832
primes of bad reduction: 2, 3, 2777, 311341
regulator: 273.08550765251130993828811941017090645271641479597
submitted by: David Renshaw
submitted at: 2026-05-28 05:01:18 UTC
last updated: 2026-07-01 22:59:17 UTC

Witness: 6 independent points log in to add more points →

(-59, 35)
(-58, 116)
(-57, 150)
(-53, 227)
(-40, 320)
(-32, 332)

Commentary

Lowest-conductor rank 6 curve in the Sage/Watkins elliptic-curves database: [1, 1, 0, -2582, 48720], conductor 5187563742. The submitted integral points were found by an exact search and certified independent by the site verifier.

last edited by David Renshaw at 2026-05-28 05:02:16 UTC · history