curve #6460

y² = x³ − 12984066028941316x

a-invariants: [0, 0, 0, -12984066028941316, 0]
c-invariants: [623235169389183168, 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): 207745056463061056
discriminant (Δ): 140091607961194413797493447814366293569460290719744
Faltings height: 8.3117
naive height: 122.9211
primes of bad reduction: 2, 47, 97, 12497
regulator: 734233.808991013845126603374389760433373584049808
submitted by: David Renshaw
submitted at: 2026-11-03 18:35:05 UTC
last updated: 2026-11-03 18:35:05 UTC

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

(-79734654, 726882229920)
(158127646, 1378673127600)
(-162462422946558959/6444878400, 288669209093450958298056839/517394837952000)
(742299817777/256, 639047927938010409/4096)
(922754418376176/3481, 28030391265259381220280/205379)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x, n = 113947646. 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 18:39:49 UTC · history