curve #4728

y² = x³ − 18366414469254577944100x

a-invariants: [0, 0, 0, -18366414469254577944100, 0]
c-invariants: [881587894524219741316800, 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): 293862631508073247105600
discriminant (Δ): 396509060811465164039447214329498950068472138802912426615239744000000
Faltings height: 11.8522
naive height: 165.4080
primes of bad reduction: 2, 5, 23, 41, 127, 113161
regulator: 29393654.474476138793587885824135906355079360350232
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:25 UTC
last updated: 2026-10-20 21:26:25 UTC

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

(4588738260533260/9801, 297535003425944535708100/970299)
(-2436661497635/81, 16707945924712689625/729)
(-96349391915974915/734449, 7753343677163842423079625/629422793)
(894562865423252035340/1604563249, 25953161405852169243810589139100/64273990065193)
(3731434741025281/2116, 227262338335460096407521/97336)
(392870515101972767209657260/733713640948201, 7533528587070739065115545543747395466900/19874210715696421768949)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 135522745210. 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 21:26:25 UTC · history