curve #5068

y² = x³ − 361386131716x

a-invariants: [0, 0, 0, -361386131716, 0]
c-invariants: [17346534322368, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 5782178107456
discriminant (Δ): 3020608367645775043859122836279660544
Faltings height: 5.6893
naive height: 91.4532
primes of bad reduction: 2, 17, 17681
regulator: 23787.40949199075170268742140365522244295622642
submitted by: David Renshaw
submitted at: 2026-10-21 02:01:23 UTC
last updated: 2026-10-21 02:01:23 UTC

Witness: 4 independent points log in to add more points →

(-304704, 286051920)
(638146, 171041760)
(-167281375266/332929, 44940946523076240/192100033)
(-17965328/49, 98944113864/343)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 601154. Found by a Mestre-Nagao sieve over n <= 10^7 (n = 1,2,3 mod 8), top candidates checked with ellrank.

last edited by David Renshaw at 2026-10-21 02:01:23 UTC · history