curve #5045

y² = x³ − 35292003673678921x

a-invariants: [0, 0, 0, -35292003673678921, 0]
c-invariants: [1694016176336588208, 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): 1129344117557725472
discriminant (Δ): 2813253846020425771713195150433606693631780763133504
Faltings height: 8.5616
naive height: 125.9209
primes of bad reduction: 2, 13, 137, 313, 337
regulator: 654823.959589476301079344791517812065371312452653
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:39 UTC
last updated: 2026-10-21 01:49:39 UTC

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

(-3873789412/169, 1961262738655362/2197)
(-294892637/9, 28589353943228/27)
(57925616229/289, 4859977078008780/4913)
(6353368906921/20736, 12650841181098263915/2985984)
(226180812164086637/151609969, 106711874405393842974119016/1866773548297)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 187861661. 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 01:49:39 UTC · history