curve #5048

y² = x³ − 258735871513261321x

a-invariants: [0, 0, 0, -258735871513261321, 0]
c-invariants: [12419321832636543408, 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): 8279547888424362272
discriminant (Δ): 1108536267458170159577637191328154585728313173515466304
Faltings height: 9.0597
naive height: 131.8974
primes of bad reduction: 2, 73, 137, 181, 281
regulator: 860240.250317562896676373022999180249673507959898
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:52 UTC
last updated: 2026-10-21 01:49:52 UTC

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

(-470920189, 4172522207280)
(-72148036, 4276883208570)
(-31685193536863725/1302415921, 117790133059660800841584960/47002888172969)
(4588705434949/6241, 7097661704426143620/493039)
(480070559985773/779689, 5927121344625790341468/688465387)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 508660861. 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 01:49:52 UTC · history