curve #5730

y² = x³ + 105464265721

a-invariants: [0, 0, 0, 0, 105464265721]
c-invariants: [0, -91121125582944]
rank (lower bound): ≥ 9
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1201252825159527130182828
discriminant (Δ): -4805011300638108520731312
Faltings height: 3.4149
naive height: 64.2864
primes of bad reduction: 2, 3, 223, 472933927
regulator: 720180075655.69187331172186182141543233684929834315173568
submitted by: jesper-petersen
submitted at: 2026-10-24 17:17:39 UTC
last updated: 2026-10-24 17:17:39 UTC

Witness: 9 independent points log in to add more points →

(-7588611844325/2192206041, 25963145197537280554/102641279045661)
(-85251388137/30217009, 47855901990130484/166102898473)
(1107652939335/241149841, 1684626788604042104/3744815880889)
(2581223882823/405498769, 4922386015079692184/8165528711353)
(5616563324055/668170801, 14444360485507495364/17271547035049)
(16123106275179/1335244681, 66650888458486653610/48791175888421)
(-3576522997119/2568259684, 41723248701415389665/130154264265752)
(48823838018145/12441617764, 565240487581705101007/1387762928632088)
(-1121102658492081/546069271225, 125554818438452017565428/403526079010782125)

Commentary

Constructed from a rank-four Mordell family obtained through a rational parametrization of the Burkhardt quartic. The construction parameters are (u,v,w) = (-1,2,2), with specialization t = -155/199. PARI/GP 2.17.4 computation returned matching rank bounds [9,9] and nine rational points on the original curve.

last edited by jesper-petersen at 2026-10-24 17:18:58 UTC · history