curve #4720

y² = x³ − 2330209058073121x

a-invariants: [0, 0, 0, -2330209058073121, 0]
c-invariants: [111850034787509808, 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): 74566689858339872
discriminant (Δ): 809775498984225888766576997151308626762825635904 ☆ record for congruent-number curves, rank ≥ 5
Faltings height: 7.8822 ☆ record for congruent-number curves, rank ≥ 5
naive height: 117.7678 ☆ record for congruent-number curves, rank ≥ 5
primes of bad reduction: 2, 23, 31, 79, 857
regulator: 235509.867956499719913784565167724758784039657111
submitted by: David Renshaw
submitted at: 2026-10-20 21:25:53 UTC
last updated: 2026-10-20 21:25:53 UTC

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

(1004028775/9, 28681487598160/27)
(98138033, 846458786712)
(-19513711, 195039737760)
(483891794847/7921, 206295102244711416/704969)
(-21616933609/1369, 9181708658384880/50653)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 48272239. Source: Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014/2, 63-98, doi:10.5802/pmb.9 (https://pmb.centre-mersenne.org/item/10.5802/pmb.9.pdf), Table 2 'Smallest rank r quadratic twists y^2=x^3-d^2x'.

last edited by David Renshaw at 2026-10-20 21:25:53 UTC · history