curve #5044

y² = x³ − 6472509822648100x

a-invariants: [0, 0, 0, -6472509822648100, 0]
c-invariants: [310680471487108800, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 103560157162369600
discriminant (Δ): 17353941477640973293849149577110961666729024000000
Faltings height: 8.1376
naive height: 120.8326
primes of bad reduction: 2, 5, 7, 11, 163, 641
regulator: 417973.406936928748807100151412503434112170190235
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:35 UTC
last updated: 2026-10-21 01:49:35 UTC

Witness: 5 independent points log in to add more points →

(216558846, 2958790267944)
(-302706147590/5329, 167040440780228400/389017)
(-1167556775/36, 90566556004925/216)
(95759668690/1089, 11961091313649800/35937)
(276202567504710/3243601, 1504112313799692735600/5841725401)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 80451910. Found by a Mestre-Nagao sieve over n <= 3*10^9 (n = 5,6,7 mod 8), top candidates checked with ellrank.

last edited by David Renshaw at 2026-10-21 01:49:36 UTC · history