curve #5042

y² = x³ − 4323110629133956x

a-invariants: [0, 0, 0, -4323110629133956, 0]
c-invariants: [207509310198429888, 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): 69169770066143296 ☆ record for congruent-number curves, rank ≥ 5
discriminant (Δ): 5170934326187742978651838259482225840884363956224
Faltings height: 8.0367
naive height: 119.6219
primes of bad reduction: 2, 11, 29, 257, 401
regulator: 101933.752901090032038746570907439878171150273527
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:27 UTC
last updated: 2026-10-21 01:49:27 UTC

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

(-45344052, 320618490348)
(-36443198, 330374820336)
(3413280222, 199378041340896)
(5346849277/81, 34674734198951/729)
(71313900445150/986049, 250845325472752024960/979146657)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 65750366. 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 01:49:27 UTC · history