curve #5523

y² + xy + y = x³ + x² − 3330505391033949539x + 2339095637981749092628899656

a-invariants: [1, 1, 1, -3330505391033949539, 2339095637981749092628899656]
c-invariants: [159864258769629577873, -2020978632415213156803591136985]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 112418315816996811
discriminant (Δ): 711390425196999918854670326027990472857843904654716289
Faltings height: 9.3388
naive height: 139.5626
primes of bad reduction: 3, 23, 29, 41, 43, 59, 83, 2297, 2833
regulator: 2465758.4720631302023779850496382524067472374779026
submitted by: Daniel Błażewicz
submitted at: 2026-10-22 21:26:33 UTC
last updated: 2026-10-22 21:26:33 UTC

(-13228834772/9, 1720160072542309/27)
(1585134316297/1521, 14864025416957725/59319)
(73540546555761/105625, 649292088000796966159/34328125)
(1427868142805662/4076361, 286931220801526829786510/8230172859)
(5628384467604642/5396329, 867211428010627955077/12535672267)
(7706511318330588/4507129, 387884308898568510592364/9568634867)

(u,v) = (9188,112631) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 6.

last edited by Daniel Błażewicz at 2026-10-22 21:26:34 UTC · history