curve #6459

y² = x³ − 9804139343349316x

a-invariants: [0, 0, 0, -9804139343349316, 0]
c-invariants: [470598688480767168, 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): 156866229493589056
discriminant (Δ): 60312648410952447921086153680271701778992178335744
Faltings height: 8.2414
naive height: 122.0783
primes of bad reduction: 2, 7, 17, 97, 4289
regulator: 177073.110113382910520043693895151531389439756108
submitted by: David Renshaw
submitted at: 2026-11-03 18:35:00 UTC
last updated: 2026-11-03 18:35:00 UTC

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

(-66389904, 598560309480)
(-245344548862/3481, 119911417411211328/205379)
(-886120802/9, 2812261580128/27)
(-725662238/9, 13933795025024/27)
(3065659198/9, 162411001589872/27)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x, n = 99015854. Found in a local squarefree-parameter search using Monsky 2-Selmer matrices and Mestre-Nagao scoring, followed by PARI/GP ellrank point search. Search performed 2026-11-03. Method reference: Dujella, Janfada, Salami (2009), https://web.math.pmf.unizg.hr/~duje/pdf/Sajad_JIS.pdf.

last edited by David Renshaw at 2026-11-03 18:39:47 UTC · history