curve #4726

y² = x³ − 15412730566502640613284x

a-invariants: [0, 0, 0, -15412730566502640613284, 0]
c-invariants: [739811067192126749437632, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 246603689064042249812544
discriminant (Δ): 234325058113846545520368187605974516981426632957152094385667666579456
Faltings height: 11.8084
naive height: 164.8820
primes of bad reduction: 2, 3, 7, 17, 41, 751, 5647
regulator: 19601741.275631068224865671682162798833186232181234
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:17 UTC
last updated: 2026-10-20 21:26:17 UTC

Witness: 6 independent points log in to add more points →

(15630483860187756/105625, 1063478409088682202316596/34328125)
(-28276242435587762/346921, 5462956723015935950708440/204336469)
(-89890066408315782/786769, 11457315952350802921094400/697864103)
(-352286054891120532/4583881, 265269359080039106910974220/9814089221)
(-151105201632378829542/59064066961, 90117807456685802222515310970480/14354399257598791)
(-170088198869948570617350/2309843713489, 95216429170143567217978852044143520/3510539743103711513)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 124148018778. Source: Watkins et al. electronic data http://magma.maths.usyd.edu.au/~watkins/PTS.r6r7 (paper Sec. 15); 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).

last edited by David Renshaw at 2026-10-20 21:26:17 UTC · history