curve #5046

y² = x³ − 85697971655981281x

a-invariants: [0, 0, 0, -85697971655981281, 0]
c-invariants: [4113502639487101488, 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): 2742335092991400992
discriminant (Δ): 40280198566442308430451132553855809873979205914306624
Faltings height: 8.7834
naive height: 128.5824
primes of bad reduction: 2, 17, 71, 409, 593
regulator: 467514.134224753821199928201219532463577288881167
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:43 UTC
last updated: 2026-10-21 01:49:43 UTC

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

(-56064813247/1024, 69726641521539297/32768)
(-16354561527/121, 4018177661997672/1331)
(3230530672/9, 106254576700084/27)
(27091467173632/91809, 17733982433256526096/27818127)
(3641430801144361/7617600, 173715534118404406897109/21024576000)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 292742159. 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 01:49:43 UTC · history