curve #5047

y² = x³ − 229696909049838916x

a-invariants: [0, 0, 0, -229696909049838916, 0]
c-invariants: [11025451634392267968, 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): 3675150544797422656
discriminant (Δ): 775613620775158995702487560609235182918025849558994944
Faltings height: 9.0299
naive height: 131.5402
primes of bad reduction: 2, 7, 23, 151, 9857
regulator: 368051.804231135245655208194504322963425713184780
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:47 UTC
last updated: 2026-10-21 01:49:47 UTC

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

(-1492638467499294687698/3347842343521, 22741504951962982371371402574336/6125583962206152431)
(-184893561160336/395641, 571945303534098900840/248858189)
(-880665975/16, 226058090411685/64)
(2080608313/4, 36881715664563/8)
(209577491876025/306916, 2161160793022671900915/170031464)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 479267054. 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 01:49:47 UTC · history