curve #5041

y² = x³ − 3448707957941401x

a-invariants: [0, 0, 0, -3448707957941401, 0]
c-invariants: [165537981981187248, 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): 110358654654124832
discriminant (Δ): 2625120427778747314905924919043896314324461068864
Faltings height: 7.9802
naive height: 118.9439
primes of bad reduction: 2, 17, 47, 67, 1097
regulator: 206653.288915877337875015182657322714044926674948
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:24 UTC
last updated: 2026-10-21 01:49:24 UTC

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

(-43633683, 259626145164)
(123286499, 1203628477380)
(3253786876, 185572196586810)
(-45852513424/37249, 468305188775130300/7189057)
(-1427932944/25, 12895067445204/125)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 58725701. 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 01:49:24 UTC · history